


A HANDBOOK 



ON THE 



TEETH OF GEARS, 



THEIR CURVES, PROPERTIES. 



AND 



PRACTICAL CONSTRUCTION. 



By GEORGE B. GRANT, M.E. 




ADVERTISEMENT OF THE 

LEXINGTON GEAR WORKS, 

GEO. B, GRANT, Proprietor. 



GEAR WHEELS 



AND 



GEAR CUTTING 



OF EVERY DESCRIPTION, 



\A/E wish to send our new 1890 GEAR BOOK to every manufacturing 
and nnechanical concern in the country. We do not distribute these 
valuable pamphlets broadcast, and cast most of them away, but when one 
is sent for it is evidence enough that it is wanted. We charge ten cents 
to parties not in business, and refund with first order. 



LEXINGTON GEAR WORKS, 

MASS. 



LIBRARY OF CONGRESS. 

: ©ujnjnjigt Ifn* 

Slielf:Afi::l-6 



UNITED STATES OF AMERICA. 



A HANDBOOK 



TEETH OF GEAES, 

THEIR OUEYES, PEOPEETIES, AKD 
PRACTICAL CONSTRUCTIQ]^ 



George B. Geai^t, M. E. 



THIRD EDITION. 
Copyright, 1890, by Geo. B. Grant. 




PUBLISHED BY THE 

LEXINGTON GEAR WORKS, 

Lexington, Mass. 



3 



\'S 



'^ 



"k^ 



.?^a®fe5(^?_. 



PRESS OF 

C. A. PINKHAM A CO. 

»89 CONGRESS ST., 

BOSTON. 



^"-^^^^^^^(^^'^ 




THE TEETH OF 

GEAR WHEELS. 



INTRODUCTION. 

Few mechanical subjects have attracted the attention of scientific men 
to such an extent, or are so intimately connected with mathematics, as the 
proper construction of the teeth of gear wheels, and, as a consequence, few 
can show such an advance as has here been made, from the rough cog 
wheel of not many years ago, to the neat cut gear of the present day. 

It is not apparent wherein much further improvement is needed in our 
knowledge of the theory of the subject, but it is evident that much remains 
to be done towards its practical application, and to induce the working 
mechanic to understand and use the improvements that have been developed 
by the mathematician and the inventor. The theory seems to be full and 
well nigh perfect, but the mill-wright and the machinist still clings to 
imperfect rules and clumsy devices that should have been forgotten years 
ago, and few workmen have a clear knowledge of even the rudiments of the 
science which it is their business to apply to practical purposes. 

It is the mathematical and scientific character of the subject that makes 
it so difficult to the practical man, who can understand but little of it as it 
is commonly presented in elaborate treatises or encyclopaedias, and who 
takes but little interest in the study of a matter that bristles with strange 
characters and technical terms. 

I have here undertaken to address the workman as well as the man of 
science, and have felt obliged to leave out nearly everything that cannot be 
treated in a plain, descriptive manner, to use language that any intelligent 
man can understand, and to refer to more pretentious works than this for 
demonstrations, or unessential details. 

A volume of a thousand pages would not properly present the whole 
subject, and this little pamphlet can deal only with the main principles and 
prominent points. It is not a treatise, it is a hand-book that does not 
pretend to cover the whole ground, and its principal object is to present 
the new odontographs, which I believe to be superior to those heretofore 
in use for the purpose of designing the teeth of gear wheels. 

FIRST PRINCIPLES. 

The original gear wheel had pins or projections for 
teeth, of any form that would serve the general purpose 
and communicate an unsteady motion from one wheel to 
another. 




The perfect gear wheel is the friction wheel, communicating 
a smooth, uniform, rolling motion, by means of the frictional 
icontact of its surface. It is, in fact, a gear wheel with a 
'great many very small, weak, nnd irregular teeth. 
The whole aim and object of the science of the teeth of gear 
fic.2. FRICTION wHEELs.wheelsisto increase the size and strength of these teeth with- 
out destroying the uniformity of the motion they transmit, 
and this is accomplished by studying the shape of the teeth, and giving their 
bearing surfaces the curved outline that is found to produce the desired 
result. 

There are an infinite number of curves that will meet the requirement, 
but only two, the epicycloid and the involute, are of any practical impor- 
tance, or in actual use. 



THE EPICYCLOIDAL TOOTH. 

The epicycloidal or double curve tooth 
has its bearing surface formed of two 
curves, meeting at the pitch line P, 
which corresponds to the working ch- 
cle of the perfect gear wheel of fig. 2. 

If a small circle,a,be rolled around on 

the outside of the pitch circle, p, a fixed 

^ tracing point, a, in its edge, will trace 

• out the dotted line called an epicycloid, 

{ and a small part of this curve near the 

I pitch line, usually one sixth of its full 

.;j)»» height,f orms the face of the tooth. 

j^y Similarly, if a small circle, B, be rolled 

around on the inside of the pitch line, 

its tracing point, b, will describe the 

internal epicycloid, or hypocycloid, a 

small portion of which is used for the 

fio.». THE ep.cyc'loidal TOOTH. flank of thc tootli. 




.0^ 




ICYCLOIOAL TEETH. 



If a projection be formed on the friction 
wheel fig. 4, the curved outline of which 
is a whole epicycloid E, and a depression 
be formed in the wheel N having a whole 
hypocycloid H for its outline, then, if both 
curves have been formed by the same 
describing circle B, it can be mathemati- 
cally demonstrated that the two curves 
will just touch and slide on each other, 
without separating or intersecting, while 
the two friction wheels roll together. 

The reverse of this fact is also true, that, 
if one wheel drives another by means of 
an epicycloidal projection on it working 
against a hypocycloidal depression in the 
other, both curves being formed by the 
same describing circle, the t-wo wheels will 
roll together as uniformly as if driven by 
frictional contact, and it is this peculiar property of the epicycloid that gives 
it its value for the purpose in hand. 

The pressure acting between the two curves is in the direction of the 
line dg, is direct only at the start, and becomes more and more oblique, 
until, when the middle points, q q, come together, and beyond, there is no 
driving action at all. This defect forbids the use of the whole curve and we 
can use but a small portion of it near the pitch line. Another projection and 
depression must be formed so near the first that they will come into work- 
ing position before the first pair are out of contact, thus forming the theo- 
retically perfect but incomplete gears of fig. 5. 

Practical requirements still further 
modify the apparent shape of the 
tooth, for it is desirable that the 
wheels shall work in either direction, 
and that they shall be interchangea- 
ble, so that any one of a set of several 
shall work with any other of that set. 
This can be accomplished only by 
making the curves face both ways, 
and by putting both projections and 
depressions on each gear, thus form- 
ing the familiar tooth of fig. 3. 




no. B. INCOMPLETE EPICYCLOIDAL TEETH. 



THE INTERCHANGEABLE SET. 

If all the curves of a set of several gears, both the faces and the flanks 
of each gear, are described by the same rolling circle, the set will be 
interchangeable, and any one will work perfectly with any other. 

This is a property of the greatest practical importance, and interchangea- 
ble sets should come into as universal use on heavy mill work as with cut p-ear- 
ing. It is the only system that will allow the use of a set of ready made 
cutters, and is therefore essential to the economical manufacture of cut 
gear wiieels. 

The diameter of the rolling circle is usually made half the diameter of the 
smallest gear of the set, and that gear will have straight radial lines for 
flanks. 

The set in almost universal use and adopted for all the odontographs, has 
twelve teeth in its smallest gear, but there is a tendency to change this well 
established system, and create confusion for which the writer can see no 
adequate excuse, by the adoption of a pinion of fifteen teeth as the base 
or smallest gear. It may be admitted that as large a base as possible should 
be used, but the change from twelve to fifteen seems to be unwarranted 
in view of the confusion it creates by the abrupt change from an old and 
good rule to a new one that is a mere shade better, and the trouble it makes 
wdth small pinions of eight to twelve teeth. 

RADIAL FLANK TEETH. 

If the internal curves, or flanks, of a pair of gears that are to run together 
are on each radial straight lines described by a rolling circle of half its 
pitch diameter, and the rolling circle that describes the flanks of one gear 
is used to describe the faces of the other gear, then, the two gears will form 
a pair fitted to each other and not interchangeable with other gears. 

This style of gear is very often used under the erroneous impression that 
it is the best possible form, and will give the least possible friction and 
thrust on the bearings, but the saving in friction over the interchangeable 
form would be an exceedingly difficult thing to measure by any practicable 
method, although it can be mathematically demonstrated to be a fact, and 
the slender roots of such teeth make them weaker and much inferior to the 
others. The odontograph figures show both a pair of these gears, and the 
same pair on the interchangeable plan, also, by the dotted lines on the former 
figure, the shapes as they would be on the interchangeable plan. It is plainly 
seen that the interchangeable faces are but a shade more rounding, while 
their flanks are so curved that the teeth are much stronger at the roots. The 
larger the describing circle, the less the theoretical thrust and friction, and 
if the flanks Avere formed by a describing circle of more than half the diam- 
eter of the gear, the teeth would be undercurved, the friction less, and their 
strength less, than that of the radial flank tooth. 

In practical matters it is a good plan to give first place to practical points, 
and not to take too much notice of minute theoretical advantages, and 
there is no good reason, that will bear the test of experiment, for adopting 
the radial flank, non-interchangeable, and weak tooth, in preference to the 
strong tooth of the interchangeable system. 

THE PITCH. 

The pitch is a term used to designate the size of the tooth, and is either 
circular or diametral. 

THE CIRCULAR PITCH or more properly the circumferential pitch, 
is the actual distance from tooth to tooth measured along the curve of the 
pitch line, and is expressed in inches, as | inch pitch, 1^ inch pitch, etc. 

The table gives the proper pitch diameter of a gear of any given number 
of teeth, and one inch circular pitch. The tabular numbers must be multi- 
plied by any other pitch that is in use. 

Formerly, the circular pitch was the only one known, but it has deser- 
vedly gone out of use on cut gears, and it is hoped may soon be abandoned 
altogether. It is a clumsy, awkward, and troublesome device on either large 
or small Avork, having its origin in the ignorance of the past, and owing its 



existence not to any perceptible merit, but to habit, anci the natural per- 
sistence of an established custom. 

With the circular pitch the relation between the pitch diameter of the 
gear, and the number of teeth on it, is fractional. If the diameter is a 
convenient quantity, such as a whole number of inches, the pitch must be 
an inconvenient fraction, and if the pitch is a handy part of an inch, the 
diameter will contain an unhandy decimal. 

With the circular pitch there is no one length of tooth that is better than 
any other, and consequently there is no agreement upon that point. Each 
maker is at liberty to chose his own distance at random, and whatever he 
choses is as good as any other. 

Its worst feature is that it leads to endless errors, for the average mechanic 
appreciates convenience more than accuracy, and will stretch his figures to 
suit his facts, with a botch as the common result. 

A millwright figures out a diameter of 22.29 inches for a gear of one inch 
pitch and 70 teeth, and failing to make such a clumsy figure fit his work or 
his foot rule, and thinking a quarter of an inch or so to be of no importance, 
he lets it go at 22 whole inches. The same process on its mate of 15 teeth 
gives a 5 inch gear instead of one of 4.78 inches diameter, and the pair will 
never run or wear together properly. His only alternatives are to adopt 
the clumsy true diameters, or else use the clumsy figure .988 inch for his 
pitch. 

Again, he is apt to apply a carpenter's rule directly to the teeth of the 
gear he is to repair or match, and naturally takes the nearest convenient 
fraction of an inch as his measurement, when the real pitch may be just 
enough diiferent to spoil the job. 

There is no reason whatever for using the circular i^itch, unless the work 
to be done is to match work already in use. 

THE DIAMETRAL PITCH is an immense improvement on the old 
fashioned circular pitch. It is not a measurement, but a number, or ratio. 
It is the number of teeth on the gear, for each inch of its pitch diameter, 
and its merit is that it establishes a convenient and manageable relation 
between these two principal elements, so that the calculations are of the 
simplest description and the results convenient and accurate. 

The product of the pitch and the pitch diameter is equal to the number of 
teeth, and the number of teeth divided by the pitch is equal to the pitch diame- 
ter. A gear of 15 inches diameter and 2 pitch has 30 teeth, and a gear of 27 
teeth of 4 pitch has a pitch diameter of 6f inches. 

The rule that the length of the tooth is two pitch parts of an inch, | or ^ 
an inch for 4 pitch, f or 1 inch for 2 pitch, etc. is so simple and so much bet- 
ter than any other that it is never disputed, and is in universal use. 

The circular and diametral pitches arc connected by the relation 

cXp=8.141G. 
or, the product of the circular and the diametral i)itch is the number 3. 1416. 

THE ADDENDUM. 

^or reasons expressed above we can use but a small part of the epicy- 
cloidal curve near the pitch line, limiting it by a circle drawn at a distance 
inside or outside of the pitch line called the addendum. The outside limit 
need not be the same as the inside limit, but it is customary to make them 
equal. 

When the diametral pitch is used, the length of the addendum is always 
one pitch part of an inch, as Jthincli for 4 pitch, ird inch for 3 pitch, etc. If 
we use the same proportion for circular pitches the addendum will be 3 xir^ 
circular pitch, and the value ^rd of the circular pitch may be adopted as the 
most convenient for use. 

THE CLEARANCE. 

Theoretically, the depression formed inside the pitch line should be only as 
deep as the projection outside of it is high, but to allow for practical defects 
in the making or in the adjustment of the teeth, and to provide a place for 



dirt to lodge, the depression is always deeper than theory requires by an 
amount called the clearance. The amount of the clearance is arbitrary, but 
the sixteenth part of the depth of the tooth is a convenient and customary 
measure, or ^^:fth of the circular pitch, and 1 divided by 8 times the diametral 
pitch. The following tables will be convenient and save calculation : 

CLEARANCE FOR CIRCULAR PITCHES. 



Circular pitch. 
Clearance. 

CLE/ 


.02 


f 
.03 


f 
.03 


i 
.04 


1 

.04 


.05 


H 

.05 


If 
.06 


.06 


If 

.07 


2 
.08 


.09 


.10 


.12 


FRANCE FOR DIAMETRAL PITCHES. 


Diametral pitch. 
Clearance. 


6 
.02 


5 
.03 


4 
.03 


H 

.04 


.04 


3 
.04 


2| 
.05 


2i 
.05 


2i 
.06 


2 
.06 


If 

.08 


H 

.09 


.10 


1 

.12 

1 



THE BACKLASH. 

When wooden cogs or rough cast teeth are used, the inevitable irregular- 
ities require that the teeth should not pretend to fit closely, but that the 
spaces should be larger than the teeth by an amount called the backlash. 
The amount of the backlash is arbitrary, but it is customary to make it 
about equal to the clearance. 

Cut gears should have no allowance for backlash, and involute teeth need 
less backlash than epicycloidal teeth. 



PITCH DIAMETERS. 



waiti ONE I^^^C]EI circxjil.^1?, jpitch. 

For Any Other Pitch, Multiply by that Pitch. 



T. 


P.D. 


T. 


P.D. 


T. 


P.D. 


T. 


P.D. 


10 


3.18 


1 

33 


10.50 


56 


17.83 


79 


25.15 


11 


3.50 


34 


10.82 


57 


18.15 


i 80 


25.47 


12 


3 82 


35 


11.14 


58 


18.47 


81 


25.79 


13 


4.14 


36 


11.46 


59 


18.78 


82 


26.10 


U 


4 46 


37 


11.78 


60 


19.10 


83 


26.43 


15 


4.78 


88 


12.10 


61 


19.42 


84 


26.74 
27.06 


16 


5.09 


39 


12.42 


62 


19.74 


85 


17 


5.40 


40 


12.74 


63 


20.06 


86 


27.38 


18 


5.73 


41 


13.05 


64 


20.38 


87 


27.70 


19 


6.05 


42 


13.37 


65 


20.63 


88 


28.02 


20 


6.37 


43 


13.69 


66 


21.02 


89 


28.34 


21 


6 69 


44 


14.00 


61 


21.33 


90 


28.65 


22 


7.00 


45 


14.33 


68 


21.65 


91 


28.97 


23 


7.32 


46 


11.65 


69 


21.97 


92 


29.29 


24 


7.64 


47 


14.96 


70 


22.29 


93 


29.60 


25 


7.96 


48 


15.28 


71 


22.60 


94 


29.93 


26 


8.28 


49 


15.60 


72 


22.92 


95 


30.25 


27 


8.60 


50 


15.92 


73 


23.24 


96 


30.56 


28 


8.90 


51 


16.24 


74 


23.56 


97 


30.88 


29 


9.23 


52 


16 56 


75 


23.88 


98 


31.20 


30 


9.55 


63 


16.87 


76 


24.20 


99 


31.52 


31 


9.87 


54 


17.19 


77 


24.52 


100 


31.84 


32 

i 


10.19 


55 


17.52 
L 


78 


24.83 

( 








The Epicycloid. 




THE EPICYCLOID. 

THEORETICAL FORMATION. 

Tlie true epicycloid, shown by fig. 6, 
is perpendicular to the pitch line at 
the origin a, and forms an endless 
series of lobes about it, as in fig. 3. 

The most convenient and simple 
process for drawing it, is to step it ofit' 
with the dividers. Several describing 
circles, M^ to M% are drawn at ran- 
dom ; steps are made, as shown by 
the figure, from the origin a^ to past 
each tangent point, a^ to a^, and then 
the same number back, around each 
circle, to locate the several points, b^ 
to b^, on the curve, which is then 
drawn by hand through the points, 
and is accurately in place if the steps 
are small. 
By the mechanical method for drawing the 
curve, the describing circle, B, is rolled around 
the pitch circle A, and a tracing point or pencil 
P, draws the curve. A steel ribbon s, is fastened 
to the templets at each end, and assists in keep- 
ing them in place. 

This process is the main principle of the epicy- 
cloidal engine, which carries a scribing tool, or 
a rotary cutter at p, to trace or cut out a tem- 
plet that is then used in forming gear teeth or 
gear cutters. 

It is, of course, the most accurate method 
known, but it is not available for ordinary pur- 
poses, for unless the templets are well made and 
skillfully handled, the resulting curve will be 
poorly drawn, and the method, although simple in principle, may be consid- 
ered difficult in its practical application. 

PRACTICAL FORMATION. 

Of course nothing but the perfect curve will answer its purpose with per- 
fect accuracy, but the epicycloid is a peculiar curve which cannot be accu- 
rately drawn by any simple process, or with common instruments, particu- 
larly when the teeth are small, and it is customary to use arcs of circles or 
other curves, which approximate as nearly as possible to the true curve. 

Such an arc can be made to agree with the curve so closely that it is a need- 
less refinement to be more particular for most practical purposes, such as 
drafting teeth, making wooden cogs or patterns for cast teeth, or even the 
templets for shaping gear cutters and planing bevel gear teeth. 

Some makers of rough cast or heavy planed gearing go to great expense 
to construct the (supposed to be) theoretically true epicycloid, by means of 
rolling circles. This practice looks very much indeed like accuracy, but if 
he had an absolutely true curve as a templet, supposing he could make such 
a thing, the maker of this class of work could not produce from it a work- 
ing tooth more nearly perfect than if the templet was properly constructed 
of circular arcs. It is labor lost to lay out teeth to the thousandth of an 
inch, that must be constructed with ordinary hand or machine tools, or 
shaped with a chisel and mallet. 

Furthermore, it is a question if the delicate processes and epicycloidal 
engines used for the finest cut gear work, can serve practical purposes and 
construct templets to work from, better than intelligent and skillful 
hand-work. It is a fact that the best work in this line is made from tem- 
plets that are laid out by theory, but dressed into shape and perfected by 
hand and eye processes. 



Fig. 7. Epicycloidal Engine. 



ODONTOGRAPHS. 

Many arbitrary or "rule of thumb" methods for shaping gear teeth have 
been propo ed, but they are generally worthless, and reliance should be placed 
only on such as are founded on the mathematical principles of the curve to 
be imitated. Of these only three are known to the writer. 

y ;>"' __.«i:t \ ^ c'^ 

J,--;--^' ^ -X ^ H" ctrei/lar pltth 

:'** / \\ c*/, 

\ 



THE WILLIS ODONTOGRAPH is a method for finding the 
center m of the circle which is tangent to the epicycloid a b c, at the point b, 
where it is cut by a line bm, which passes through the adjacent pitch point 
k, and makes the angle gkf=75^ with the radial line kf. 

The radius used, is not the line m b, but the more convenient line m a. 

The instrument is nothing whatever but a piece of card or sheet metal 
cut to the angle of 75", which is laid against the radial line kf, as a guide 
for drawing the line km. The center distance km, to be laid off along the 
line thus draAvn is given by a table that accompanies the instrument. 

'^o instrument is necessary, for the line k m may be placed by drawing the 
arc f g with a radius of one inch, and laying off the chord f g=1.22 inch. The 
tabular distance k m can be readily computed from 



k, m, =: 



ko m. 






in which c is the circular pitch in inches, and t is the number of teeth in 
the gear. 

The Willis odontograph, as found in use, is confined to the single case of 
an interchangeable series running from twelve teeth to a rack, but for any 
possible pair of gears the angle becomes 

g k f = 90" — 1^ 
s 



and 



k, m, =: ^ . 4_ . sin. 180! 
' 6.28 t + s s 

6.28 t — s s 

in which t is the number of teeth in the gear being drawn and s the number 
in the mate. 

The accuracy of the Willis circular arc will be examined further on. 



THE IMPROVED WILLIS ODONTOGRAPH. 

EPICYCLOIDAL TEETH. 
TWELVE TO RACK. INTERCHANGEABLE SERIES. 







FOR ONE 




For ONE INCH 


NUMBER OF 


DIAMETRAL PITCH. 


CIRCULAR PITCH. 


TEETH 


For any other pitch, 


divide 


For any other pitch, mul- 


IN THE QBAB. 


by that pitch. 




tiply by that pitch. 






Faces. 


Flanks. 


Faces. 


Flanks. 1 


Exact. 
12 


Intervals. 


Rad. 


Dis. 
.15 


Rad. 


Dis. 

OO 


Rad. 

.73 


Dis. 


Rad. 


Dis. 


12 


2.30 


oo 


.05 


OO 


OO 


13i 


13-14 


2.35 


.16 


15.42 


10,25 


.75 


.05 


4.92 


3.26 


15i 


15-16 


2.40 


.17 


8.38 


3.86 


.77 


.05 


2.66 


1.24 


m 


17-18 


2.45 


. .18 


6.43 


2.35 


.78 


.06 


2.05 


.75 


20 


19-21 


2.50 


.19 


5.38 


1.62 


.80 


.06 


1.72 


.52 


23 


22-24 


2.55 


.21 


4.75 


1.23 


.81 


.07 


1.52 


.39 


27 


25-29 


2.61 


.23 


4.31 


.98 


.83 


.07 


1.36 


.31 


33 


30-36 


2.68 


.25 


3.97 


.79 


.85 


.08 


1.26 


.26 


42 


37-48 


2.75 


.27 


3.69 


.66 


.88 


.09 


1.18 


.21 


58 


49-72 


2.83 


.30 


3.49 


.57 


.90 


.10 


1.10 


.18 


97 


73-144 


2.93 


.33 


3.30 


.49 


.93 


.11 


1.05 


.15 


290 


145-rack. 


3.04 


.37 


3.18 


.42 


.97 


.12 


1.01 


.13 



THE IMPROVED WILLIS ODONTOGRAPH. 

I have carefully calculated the distances nii Uj and nig Ug of the circles of 
centers from the pitch line, and also the radii ai m^ and ag ni2, and have 
arranged them in the table above, so that the data resulting from the usual 
process can be obtained without the usual labor. 

This improved Willis process will produce exactly the same circular arc 
as the usual method, with the same theoretical error, but its operation is 
simpler and less liable to errors of manipulation. 

By the usual process it is necessary to draw two radial lines, and to lay off 
a line at an angle with each. The tabular distances laid off on these lines, 
will locate the two centers. The two circles of centers are then drawn 
through them, and the dividers set to the radii to be used. 

By the new process the circles of centers are drawn at once without pre- 
liminary constructions, at the tabular distances from the pitch line, and the 
table also gives the radii to be taken on the dividers. No special instru- 
ment is required, no angles or special lines are drawn to locate the centers, 
and the chance of error is much less. 

This process, however, is not as correct, and is no simpler or more con- 
venient than the new odontographic process given further on. 




ROBINSON'S TEMPLET ODONTOGRAPH. 

This ingenious instrument, the invention of Prof. S. W. Eobinson of the 
Ohio State University at Columbus, is based on the fact that some part of 
a certain curve of uniformly increasing curvature, called the logarithmic 
spiral, can be made to agree with the true curve of a gear tooth with a degree 
of approximation that is very precise. 

It is a sheet metal templet having a graduated curved edge a c, shaped to 
a logarithmic spiral, and a hollow edge a b shaped to its evolute, an equal 
logarithmic spiral. 

To apply the instrument, draw a radial line from the pitch point d on 
the pitch line, and another from e, the center of the tooth, and then draw 
tangents d g and n e f , square with the radial lines. 

The instrument is then so placed that a certain graduation, given by 
accompanying tables, is at the point h on the tangent nef, while the. grad- 
uated edge ac, is at the pitch point d,and the hollow edge ab, just touches 
the tangent line nef at k, and then the face of the tooth is drawn with a pen 
along the graduated edge. The flank is similarly located by placing the 
instrument so that a certain other graduation is at the pitch point d, while 
its hollow edge touches the tangent line g d. 

The full theory of this instrument would be out of place here, but may be 
found in No. 24 of Van ISTostrand's Science Series, or in Van Nostrand's Mag- 
azine for July, 1876. 



A NEW ODONTOGRAPH. 

Having frequently to apply the "Willis Odontograpli, it occurred to me 
that the process would be much simplified and much time and labor saved 
if the location of the circles of centers and the lengths of the radii were 
computed and tabulated, thus forming the improved Willis method already 
described. 

It was then evident that the process would be precisely the same, and the 
result much improved, if the centers tabulated were the centers of the near- 
est possible approximating circles, rather than of the Willis circles, and 1 
have embodied this idea in the following tables. 

I have carefully computed, by accurate trigonometrical methods, and have 
tabulated the location of the center of tlie circular arc that passes through 
the three most important points on the curve, at the pitch line a, fig. 9, 
at the addendum line k, and the point e, half way between. 

The tables locate this center directly, giving its distance from the pitch 
line, and from the pitch point. 

The circles of centers are drawn at the tabular distances " dis" inside and 
outside the pitch lines, and all the faces and flanks are drawn from centers 
on these circles, with the dividers set to the tabular radii "rad." 

The tables are arranged in an equidistant series of twelve intervals. For 
ordinary purposes the tabular value for any interval can be used for any 
tooth in that interval, but for greater precision it is exact only for the 
given "exact" number, and intermediate values must be taken for inter- 
mediate teeth. 

The tables are arranged for both the diametral and circular pitch sys- 
tems. The former is much the more manageable and should be used when 
the work is not to interchange with work already made on the latter 
system. 

The first table, giving an interchangeable set, from twelve teeth upwards, 
is the one for general use. 

The second, or radial flank table, is inserted because teeth are sometimes 
drawn that way, but, as before explained, they are weak, not interchange- 
able, and but a mere shade more direct in their action than the interchange- 
able style. 

ACCURACY OF THE ODONTOGRAPH. 

The assertion is often made that no circular arc can be made to do duty for 
the epicycloid, except for rough work, but it can be shown that the state- 
ment is not true if applied to the new method, for few mechanical processes 
can be made to work closer to a given example, than this arc is close to the 
true curve. 

Figure 9 shows the true curve, and both the 
new and the Willis aj^proximating arcs, the 
actual proportions being exagerated to show the 
errors more clearly. 

The Willis arc runs altogether within the true 
curve, while the new arc crosses it twice. 

We will take, for an example, the case of a 
twelve tooth pinion, Avhich will show the errors 
at their greatest, and calculate them with great 
care for a tooth of three inch circular pitch, which 
is twice the size of the figure on page 13, and 
may be considered a very large tooth. 
y ^^~' The distance from pitch line to addendum line 

is divided into eight equal spaces by parallel cir- 
cles, and the distance along each circle, in ten thousandths of an inch, from 
the true curve to each odontographic arc, is as follows : 






GKANT. 


WILLIS. 


At a 


.0000 


.0000 inches 


" b 


+.0088 


+.0175 " 


•' c 


+.0091 


+.0244 " 


u ^ 


+.0050 


+ .0283 " 


" e 


.0000 


+.0288 '• 


U f 


-.0086 


+.0297 " 


u 


-.0061 


+.0308 " 


" h 


-.004(5 


+.0342 " 


" k 


.0000 


+.0397 " 



rt 


= 12 


a 


20 


a 


40 


<' 


100 


a 


300 



Average, .0042 .0260 " 

It is seen that the new arc is in no place one hundredth of an inch in error, 
and that for a tooth of four pitch, a large size for cut work, its average 
error is one thousandth of an inch. A greater accuracy than this would be 
of no practical value. 

The twelve tooth gear, for which the errors of both arcs were com- 
puted, shows them at their maximum value, for, as the number of teeth in 
the gear increases, the errors diminish, and for several locations their values 
for the new arc at c, which is the point of greatest error, are as follows : 

c — .009 inches. 
" .008 " 
'• .006 " 
" .004 " 
" .002 '• 
and the errors of tne Willis arc are subject to the same rule. 

The error of the Willis arc is plainly shown, at its greatest value, by the 
figure on page 13, where the dotted faces of the pinion teeth are correctly 
located by the Willis method. 

To further test the accuracy of the new method, construct the same tooth 
face several times by the same;iprocess, using either the method by points, 
or the usual Willis process. Unless the work is most carefully performed, 
it wall be found that the several results will not agree with each other by 
amounts that are noticeable, while by the new^ method they will be sub- 
stantially the same curve. 

The new arc is most nearly correct at the most important point, the 
upper part of the curve, just where the Willis arc is most out of place, or 
w^iere the true curve, unless drawn by some delicate and costly apparatus, 
in most likely to be out of place. 

CIRCULAR AND DIAMETRAL PITCHES COMPARED. 



CIR. P. 


DM. P. 


e 


.52 


^h 


.58 


5 


.68 


4i 


.70 


4 


.78 


H 


.90 


3 


1.05 


21 


1.15 


2i 


1.25 


H 


1.40 


2 


1.57 


11 


1.80 


Ih 


2.10 


n 


2.50 


1 


3.14 


1 


4.20 


i 


6.28 



DM. P. 


CIR. P. 


k 


6.28 


1 


4.20 


1 


3.14 


H 


2 50 


U 


2.10 


11 


180 


2 


1.57 


2h 


1.25 


3 


1.05 


3i 


.90 


4 


.78 


5 


.63 


6 


.52 


7 


.45 


8 


.39 


9 


.35 


10 


.31 





THE NEW ODONTOGRAPH. 







GENERAL DIRECTIONS. 

Draw the pitch line and divide it for the pitch points mag. Take from 
the tables, multiply or divide, as the case may require, by the pitch in use, 
and lay off, the addendum a b and a c, the clearance e f , the backlash g g', 
the face distance a d, and the flank distance a c. Draw the addendum line 
through b, the root line through e, the clearance line through f, the line 
of face centers through d, and the line of flank centers through c. Set the 
dividers to the face radius, and draw all the faces ab from centers A. Set 
to the flank radius, and draw all the flanks a k from centers B. Round the 
flanks into the clearance line. The flanks of a gear of twelve teeth are 
straight radial lines. 

ODONTOGRAPH TABLE. 

EPICYCLOIDAL TEETH. 

INTERCHANGEABLE SERIES. 
From a Pinion of Twelve Teeth to a Rack. 





1 




FOR ONE 


FOR ONE INCH \ 


NUMBER or 

TEETH 


DIAIV 

For ar 


[ETRAL P 


ITCH. 

vide by 


CllB 

Fori 


LCULAR PITCH. 1 


ly other pitch, di 


my other pitch, multiply 1 


IN THE GEAR. 




that pitch. 


by that pitch. | 






Faces. 


Flanks. 


Faces. 


Flanks. | 


Exact. 


Intervals. 


Rad. 


Dis. 


Rad. 


Dis. 


Rad. 


Dis. 


Rad. 


Dis. 


12 


12 


2.01 


.06 


CO 


CO 


.64 


.02 


CO 


oo 


134 


13-14 


2.04 


.07 


15.10 


9.43 


.65 


.02 


4.80 


3.00 


15h 


15-16 


2.10 


.09 


7.86 


3.46 


.67 


.03 


2.50 


1.10 


m 


17-18 


2.14 


.11 


6.13 


2.20 


.68 


.04 


1.95 


.70 


20 


19-21 


2.20 


.13 


5.12 


1.57 


.70 


.04 


1.63 


.50 


23 


22-24 


2.26 


.15 


4.50 


1.13 


.725 


.05 


1.43 


.36 


27 


25-29 


2.33 


.16 


4.10 


.96 


.74 


.05 


1.30 


.29 


33 


30-36 


2.40 


.19 


3.80 


.72 


.76 


.06 


1.20 


.23 


42 


37-48 


2 48 


.22 


3.52 


.63 


.79 


.07 


1.12 


.20 


58 


49-72 


2 60 


.25 


3 33 


.54 


.83 


.08 


1.06 


.17 


97 


73-144 


2.83 


.28 


3.14 


.44 


.90 


.09 


1.00 


.14 


290 


145-rack. 


2.92 


.31 


3.00 


.38 


.93 


.10 


.95 


.12 



A PRACTICAL EXAMPLE 
OF THE WORK OF THE NEW ODONTOGRAPH. 




Fig. 10. 



INTERCHANGEABLE SERIES. 



Example. — A gear of 24 teeth, and a gear of 12 teeth, of li circular 
pitch. 

Data. — Take from the table the numbers to be used, which are as follows 
when multiplied by li. 

For 24 teeth, face rad, = 1.08 face dis, := .07. 
" 24 " flank " — 2.15 flank '' — .54. 
" 12 " face " = .96 face '* = .03. 
" 12 " flank "=00 flank " = co 

Also take from the proper tables the pitch diameters 5.73 and 11.46 inches, 
the addendum, .5 inch, and clearance, .06 inch. 

Process. — Draw the two pitch lines, and divide for the pitch points. Draw 
the addendum, root, and clearance lines of both gears. 

Draw the circles of centers, .03 inside the pitch line of the 12 tooth gear, 
and .07 inside of that of the other. Draw the circles of flank centers, ..54 
outside the pitch line of the 24 tooth gear, and draw straight radial flanks 
for the 12 tooth gear. 

Draw the faces of the 12 tooth gear with the radius. 96, and draw the faces 
of the 24 tooth gear with the radius, 1.08, and the flanks with the radius 2.15. 

Round the flanks into the root line, and allow backlash by thinning the 
teeth according to judgement. 

The dotted faces of the 12 tooth gear show them as they would be laid 
out by the Willis odontograph, and the figure also shows the two centers 



RADIAL FLANK SYSTEM. 
TEETH NOT INTERCHANGEABLE. 

Gears on this system must work together in pairs, each gear being fitted to 
its mate and to no other. See page 3. The process is the same that has been 
described on page 12 for the interchangeable set. 




Fig. 11. 



RADIAL FLANK SYSTEM. 



ExPLAXATiox OF THE TABLE. — The uppcr number in each square is the 
face radius, the lower is the center distance. 

The centers are mostly insid the pitch line, but some are on^the line, and 
those ha^dng the negative sign are outside of it. 

The tabular numbers are for one inch circular pitch, and must be multi- 
plied by any other circular pitch in use. For the value for any diametral 
pitch, multiply the tabular number by 3.14, and then divide by the diame- 
tral pitch in use. 

Example. — A gear of 12 teeth, paired with a gear of 24 teeth. Circular 
pitch li inches. 

Data. — Take from the table for 12 teeth into 24, face radius =.68 and cen- 
ter distance = 0, and for 24 teeth into 12. radius = 72, and distance = .05. 
These multiplied by 1^ give the values for use on the drawing, 12 rad. =1.02, 
12 dis = 0, 24 rad. = 1.08, and 24 dis. = .07. 

The addendum is one third the pitch, = i inch, and the proper tables give 
the clearance =.06, and the pitch diameters = 5.73 and 11.46 inches. 

Process. — Draw the two pitch lines 5.73 and 11.46 inches in diameter and 
space them for the teeth. 

Lay ofE the addendum, .5 inch, and the clearance, .06 inch, and draw the 
addendum, root, and clearance lines. , . , i- 

Draw all the faces of the twelve tooth gear, from centers on its pitch hne, 
with the radius 1.02. Draw all the faces of the 24 tooth gear from centers 
on a line .07 inch inside its pitch line, with the radius 1.08 inches. Draw 
straight radial lines for the flanks of all the teeth. 



ODONTOGRAPH TABLE. 

EPICYCI.01DAI. TEETH. 

RADIAL FLANK TABLE. 

FOR ANY POSSIBLE PAIR OF GEARS, NOT INTERCHANGEABLE. 

Multiply by the Circular Pitch. 

Divide by the Diametral Pitch, and then multiply by 3.14. 



NUM 
TF.ETH 
BEING 

Exact. 


BER OF 
JN GEAK 
DRAWN. 

Intervals 


NUMBER OF TEETH IN THE MATE. 

,o 13 15 . 17 19 22 25 30 37 49 73 145 
^2 14 16 18 21 24 29 36 48 72 144 rack 


12 


12 


.64 
.02 


.64 
.01 


.65 
.01 


.66 
.01 


.67 



.68 



.69 
-.01 


.70 
-.01 


.71 
-.02 


.73 

-.02 


.74 
-.03 


.75 
-.03 


13^ 


13-14 


.65 
.02 


.66 
.02 


.67 
.01 


.68; .69 
.01 .01 


.70 



.72 



.74 
-.01 


.75 
-.01 


.76 
-.02 


.78 
-.02 


.79 
-.03 


15i 


15-16 


.67 
.03 


.68 
.02 


.69 
.02 


.70 ' .72 
.01 .01 


.74 
.01 


.75 



.78 



.79 
-.01 


.82 
-.02 


.84 
-.02 


.84 
-.03 


m 


17-18 


.68 
.04 


.70 
.03 


.71 
.02 


.73 i .75 
.02 .01 


.77 
.01 


.78 
.01 


.82 



.84 
-.01 


.87 
-.01 


.89 
-.02 


.90 
-.03 


20 
23 


19-21 


.70 
.04 


.72 
.04 


.74 
.03 


.76 .79 
.02 j .02 


.81 
.01 


.83 
.01 


.87 



.90 



.93 
-.01 


.96 
-.02 


-.03 


22-24 


.72 
.05 


.74 
.04 


.76 
.04 


.79 
.03 


.82 
.02 

.85 
.03 


.84 
.02 


.87 
.01 


.91 
.01 


.94 



.98 
-.01 


1.01 
-.02 


1.03 
-.03 


27 


25-29 


.74 
.05 


.76 
.05 


.79 
.04 


.82 
.04 


.87 
.02 


.92 

.02 


.96 
.01 


.99 



1.03 
-.01 


1.07 
-.02 


1.10 
-.03 


33 


30-36 


.76 
.0(i 


.79 
.05 


.83 
.05 


.86 
.04 


.90 
.03 


.94 
.03 


.98 
.02 


1.02 
.01 


1.06 



1.11 



1.17 
-.01 


1.23 
-.02 


42 


37-48 


.79 
.07 


1 
.83 .86 
.06 .05 


.90 
.05 


.96 
.04 


.98 
.04 


1.03 
.03 


1.08 ' 1.14 
.03 1 .02 


1.20 



1.25 



1.37 
-.01 


58 


49-72 


.83 
.08 


.87 
.07 


.91 
.07 


.96 
.06 


1.02 
.06 


1.05 
.05 


1.10 
.04 


1,17 1.24 
.04 .03 


1.30 
.02 


1.43 



1.58 



97 


73-144 


.90 
.09 


.93 

.08 


.97 
.08 


1.01 
.07 


1.07 
.07 


1.11 
.06 


i:i8 

.06 


1.28 1.34 
.05 .04 


1.47 
.03 


1.65 

.02 


2.03 



290 


145 rack 


.93 
.10 


.96 
.09 


1.00 
.09 


1.05 
.09 


1.10 

.08 


1.16 

.08 


1.24 
.07 


1.37 1 50 
.07 .06 


1.70 
.04 


2.12 
.03 


2.90 
.02 



THE INVOLUTE TOOTH. 




With the exception of tlie epicycloid^ the only curve in extensive use for 
the working face of a gear tooth, is the involute. 

THE INVOLUTE CURVE. 

As the rolling circle A of fig. 3 increases in size, it finally, when of infinite 
^ ^- diameter, becomes the straight line d g of fig. 

15, while the epicycloid traced by a fixed point 
in the circle becomes the involute. 

The involute is, therefore, not a new or sep- 
arate curve, but simply a particular case of the 
epicycloid. It is the infinite form of the epicy- 
cloid.* 

As the rolling circle of infinite diameter is the 
same thing as a straight line, the involute can 
be formed by a fixed tracing point in a cord 
which is unwound from a circle, called its " base 
circle," which has been wrapped or "involved" 
FIG. 15. THE INVOLUTE. 1^ ^ ^^^ from tlils propcrty i,t derives its name. 

ITS UNIFORM ACTION. 

If the two circles A and B, fig. 16, are separ- 
ated by the distance ab, and work together by 
means of two external epicycloids C and D, the 
motion communicated will be irregular, for the 
conditions of uniformity are that the two cir- 
cles shall touch, and that the external curve 
of one shall work with the internal curve of the 
other. See page 2 and figure 4. 

The amount of this irregularity will depend 
on the proportion between the separating dis- 
tance a b and the diameter of the rolling circle 
which describes the epicycloids. If the pro- 
portion is very small, the irregularity will be 
very small, and if the rolling circle has an in- 
finitely great diameter, the proportion and the 
irregularity will be infinitely small, that is, zero. Therefore, involutes 
will work together with perfect regularity and are suitable curves for gear 
teeth. 

ITS ADJUSTIBILITY. 

If the rolling circle is infinitely large, the proportion between the separat- 
ing distance and it will always be zero, and it will not be altered by any finite 
alteration of the former, and therefore the uniformity of the action of 
involute teeth is not in any way dependent upon, or affected by any change of 
the separating distance. The action will be perfect as long as the curves 
remain in contact, and this is a property of the greatest practical value, 
which gives the involute a great advantage over every other known or pos- 
sible curve. 

The curve of any gear tooth must of necessity be a " rolled curve " formed 
by a fixed object attached to the plane of or moving with some curve that 
rolls upon the base curve of the tooth, and, as the involute is the infinite 
form of any rolled curve, it is the only form that can possess this property 
of adjustibility. 

*The exact nature of the involute curve is more fully treated of in a paper in the appendix, 
on " The Normal Theory ot the Gear Tooth Curve." 




EPICYCLOIDS. 



ITS UNIFORM PRESSURE AND FRICTION. 

The point of contact of the two involutes C and D will always be upon the 
straight line of action mn, the common tangent of the two base circles, 
commencing at its point of tangeucy with one circle, and ending at the same 
point on the other. 

The direct pressure between the two teeth will always be in the direction 
of the line of action, and uniform both in direction and in amount, a prop- 
erty that is peculiar to the involute curve, and which contributes greatly to 
the smooth action and even wear of involute teeth. Friction is substan- 
tially in proportion to direct pressure, and when the pressure is uniform, 
the friction will be uniform, and no part of the curve will be more likely to 
wear away than any other part. The durability of a tooth, particularly 
when doing heavy work, depends on the uniformity of the friction as well 
as upon its absolute amount. 

THEORETICAL CONSTRUCTION. 

To draw the involute curve through the pitch point a of two pitch circles 

A and B, draw the line of action m n at any desired angle with the line of 

centers, usually 75°, and then draw 
the base circles C and D, touching the 
line of action at e and d, where the 
perpendicular radial lines e g and f d 
meet it. From a, step off any num- 
ber of . short steps along the line of 
action and around the base line to any 
point s, then draw any number of 
tangent lines b c, t v, then step olf the 
distances sbc, stv, sb, etc., each 
equal to s d a, and the points c, v, b, 
etc., will be points of the curve. Any 
line, as w c X drawn through c at right 
angles to he, will be tangent to the 
curve. The working part of the 
curve must not be extended beyond 
the circle k e p through the point of 
contact of the line of action m n and 

the base line C, for beyond that point it will interfere with the radial flank 

of the tooth it works with. 
The curve is generally limited by the addendum line z y, at an arbitrary 

distance from the pitch line B, and ends at b on the base line D, where it :» 

perpendicular to the base line. It is continued within the base line by a. 

radial line as far as the root line zy, and is then rounded into the clearance 

line. 
The matter under epicycloidal teeth, pages 3, 4, and .5, regarding the pitch, 

addendum, clearance, and backlash, will apply as well to involute teeth. 

ANGLE OF ACTION. 

The angle mag may be less, but not greater, than the value found from 
the formula 

.180° 
m a g = 90<^ — -— - 

in which s is the number of teeth in the smallest gear in the pair. If the 
angle is greater than this the motion will not be continuous, as each pinion 
tooth will pass out of action before the next one is in position to act. 

INTERCHANGEABLE SETS. 

Any number of involute gears from base circles of different diameters will 
work together correctly and interchangeably if all are of the same pitch, and 
have the same angle of action. 
If we put s = 12 teeth, we find 

ISO" 
m ag = 90° — -^ = 75" 




:ONSTRUCTICN. 



the value for the common twelve to rack interchangeable set, and if we use 
fifteen as the smallest number of teeth in the set, we have an angle of action 
of 78°. 

PRACTICAL CONSTRUCTION. 

When the involute is to be brought into use, we meet with the same diffi- 
culties as with the epicycloid, for its theoretically correct construction is 
not easily and accurately accomplished, and we must adopt some short cut 
of approximative accuracy. 

The principle of the epicycloidal engine of fig. 7 may be applied to the 
construction of the involute, the ribbon s being drawn tight and straight as 
it is unwound from the base circle, but the same difficulties prevent its use 
for ordinary purposes. 

THE OLD RULE. 

A defective rule in common use drawls the whole curve from base line to 
addendum line, as one circular arc. The angle 
mag is laid olf at 75", sometimes at Ib^^, the 
distance a c is made equal to one quarter of the 
pitch radius a g, and the tooth curve is drawn 
from c as a center. 

This rule is simple, to be sure, but it gives the 
faces shown by the dotted lines of the figure on 
page 23, and is abominably wrong and worth- 
less. 

If it would round off the points of the teeth 
of a large gear, it would be useful to correct 
interference, but it greatly rounds the teeth of 
a small gear that needs little or no correction, 
and gives the curf e on a large gear in nearly its 
theoretical position, without the allowance for 
interference that must be made. 

It is not to be wondered that the involute tooth is in small favor with 
practical mechanics who use this bungling method, and who do not under- 
stand that the trouble is not in the involute system, but in its defective 
application. 

A NEW METHOD. 

In devising a method for drafting the involute tooth, I have borne in 

mind that a minute degree of accu- 
racy is not the essential requirement, 
for although substantial accuracy 
must be secured, simplicity and con- 
venience are qualities that must also 
be considered. 

The method, in general terms, and 
given in full on pages 22 and 23, is to 
give, by a table, the distance of the 
base circle B, see fig. 19, inside the 
pitch circle P, and to give by the same 
table, the distances or radii ac and ad 
F.c. 13. THE NBw METHoo. f^om thc pitch polut a to ccuters c and 

d on the base Ime. The face arc a w is 
drawn from the center d and the flank arc a v from the center c. 

The table, page 22, is for one diametral pitch, and covers the common 
twelve to rack interchangeable set. 




THE OLD RULE. 




INTERFERENCE. 

As indicated above, the involute face will interfere with the radial flank 
of the mating tooth if the addendum is greater than a certain amount, and 
ris the addendum in common use for the interchangeable set generally 
i^xceeds this limit, we must gererally make corrections to avoid this trouble. 




INTERFERENCE 



Interference 



Table 

7 inch cir- 



For one diametral pitch and 3 
cular pitch. Angle of action, 75°. 

Number of Teeth in the Mate. 
13 15 17 19 
14 



12 



16 18 21 



Figure 20 shows the interference, its effect, and i ts correction. 

The working' face of the involute 
should be limited at i by the circle 
k p through the tangent point e, but 
if the usual addendum continues it 
beyond that line, to s, the extension 
si will interfere with the radial flank 
c f , and the uniformity of the action 
will be destroyed. 

To correct it we must either 
weaken and spoil the shape of the 
mate tooth by undercutting the 
flank c f by an epitrochoidal line c g, 
or we may, and much better, round 
off the point of the tooth by an epi- 
cycloidal curve i h. 
The amount of this interference will depend on, and increases with, the 
angle of action, and also depends upon the number of teeth in each gear. It 
is greatest on a large gear or rack that runs in a small pinion, and least on 
a pinion running in a large gear. When the angle of action is 75° there 
is no interference when both gears of a pair have thirty or more teeth, or 

when an equal pair have twenty-one or 
more teeth. When onj gear has more, 
and the other has less than thirty teeth, 
the larger may need correction, but the 
smaller never will. 

The amount of the interference, the 
correction to be made by rounding off the 
point of the tooth, is very small and may 
generally be neglected on small pinions. 
It is given by the lower figures in the 
table, which shows that it is never more 
than a sixteenth of an inch on a large 
tooth of one diametral, or three inch 
circular pitch, and not over two or three 
hundreths of an inch on a gear of that 
pitch having few teeth. The table also 
shows by the upper figures the limit 
point or distance i x above the pitch line 
where the interference commences. 

The tabular numbers must be divided 
by the diametral pitch that maybe in use. 
and for any circular pitch it is sufficient 
to divide the tabular number by 3 and 
then multiply by the pitch. 

The table takes no notice of an inter- 
ference of less than a hundredth of an inch 
on a tooth of three inch circular pitch. 

When, as is usually and should always 
be the case, the gear being drawn belongs 
to the twelve to rack interchangeable set, 
the interference should be computed for 
a mate gear of twelve teeth , or by the first 
vertical column of the table. In this case 
the error will not be perceptible if the 
limit distance to point of first interfer- 
ence be always assumed to be half the 
addendum. 

When the work is upon a rough cog- 
wheel or mill gear, or upon a pattern for 
a cast gear, the only correction needed 
for interference, is a slight rounding olV 
of the points if it is a rack or very larger 
gear, and a mere touch on the point of a 
gear of fcAv teeth. 



12 



13-14 



I 1.5-16 



17-18 



19-21 



22-24 



c 25-29 



o 30-36 



u 37-48 

Si 

3 49-72 

2 



73-144 



145-00 



.58 
.01 

.56 
.02 

.54 

.02 

.53 
.02 

.51 

.02 

.50 
.02 

.49 
.03 

.47 

.03 

.45 
.03 

.44 

.04 

.42 
.05 

.40 

.06 


.67 
.01 

.66 
.01 

.64 
.01 

.62 
.02 

.60 
.02 

.58 
.02 

.57 

.02 

.55 
.02 

.53 

.02 

.52 

.03 

.49 
.04 

.46 
.05 


.75 

.01 

.72 
.01 

.69 
.01 

.67 
.01 

.65 
.02 

.63 
.02 

.61 
.02 

.59 
.02 

.56 
.03 

.53 
.04 


.75 
.01 

.72 
.01 

.69 
.01 

.66 
.02 

.63 
.02 

.60 
.02 


.73 

.01 

.70 
.01 

.67 
.01 



EPICYCLOIDAL vs. INVOLUTE TEETH. 

A COMPARISON. 

The epicycloidal tooth is in much greater use and favor than the involute 
form, particularly for heavy work, both writers and mechanics generally 
preferring it, and seldom giving the preference to its rival. It is difficult to 
account for this favor except, as in the case of the circular pitch system, on 
the ground that the epicycloid was adopted in the infancy of mechanical 
science, and holds its place by virtue of prior possession, for the involute 
has certainly the advantage from every practical point of view. 

Space will not permit an extended discussion with the necessarily bulky 
demonstrations, but, if the two curves be closely and carefully examined 
under the same conditions within the limits of either the twelve tooth or the 
fifteen or higher tooth interchangeable series, with the customary adden- 
dum, which limitation will cover nine-tenths of the gears in actual use, it 
will be found that they compare as follows : 

I. Adjustibility. Involute teeth alone can possess the remarkable and 
practically invaluable property, that they are not confined to any fixed 
radial position with respect to each other, for, as long as one pair of teeth 
remains in action mitil the next pair is in position, the perfect uniformity 
of the action of the curve is not imj)aired. 

The shafts may be at the proper distance apart, or not, as happens, and 
they may change position by wearing, or variably as when used on rolls, or 
may be forced together to abolish backlash, and, in fact, the curve is won- 
derfully adapted to the variable demands, and will accommodate itself to 
errors and defects that cannot be avoided in practice. 

Epicycloidal teeth must be put exactly in place and kept there, and the 
least variation in position, from bad workmanship in mounting, or by wear 
or alteration of the bearings in use, will destroy the uniformity of the 
motion they transmit. When perfectly mounted and carefully kept in 
order, epicycloidal teeth are as good as any in this respect, but for most 
practical purposes they are decidedly inferior. 

This virtue of the involute is always recognized by writers, but is seldom 
given the position its importance demands, for it is only as a result of exj^e- 
rience in making and using gears, that its importance can be seen at its full 
value. 

II. Uniformity. The direct force exerted by involute teeth on each 
other, is exactly uniform, both in direction and in amount, and this property 
ensures a uniform wearing action of the teeth, a nearly uniform thrust on 
the shaft bearings, and a steadiness and smoothness of action that cannot be 
claimed for epicycloidal teeth under any circumstances. 

The direct pressure acting between epicycloidal teeth is variable in 
amount and very variable in direction, and consequently the friction and 
wearing action between the teeth, as well as the thrust on the bearings, is 
variable between wide limits. 

III. Friction. The measure, for purposes of comparison, of the loss of 
power by friction, is the product of the direct pressure between the teeth, 
multiplied by their rate of sliding motion on each other. 

This measure is always in favor of the involute by a decided advantage, 
although the advantage is usually claimed for the epicycloid, both as to 
maximum values and average values, and as this is an important point, it 
should have great weight in deciding between the two forms of teeth, for 
the element of friction is of chief importance in determining the life of a 
gear in continual and heavy service. 

The epicycloid is mostly in use for heavy gearing from a mistaken view of 
this point, it being generally taught that its friction is the least. 

IV. Thrust ON Bearings. Here the advantage is with the epicycloidal 
tooth, but not by a large amount, and not in a matter of first consequence. 

The thrust on the bearings due to the action of the teeth on each other is 
but a fraction of the whole thrust due to the power being carried, and as 



the average thrust of the teeth is but little in favor of the epicycloid, and as 
the maximum thrust is always from that form of tooth, the two forms may 
be said to be well balanced in this respect. Moreover, the thrust of the 
involute is but slightly variable, while that of the epicycloid varies from 
large values at the points of first and final action to nothing at all at the 
line of centers, and must give rise to a rattling and uneven action. 

V. Strength. The weakest part of a tooth is at its root, and as the 
involute tooth spreads more than the epicycloidal tooth, it is stronger at that 
point and has a considerable advantage. 

YI. Appearance. This is a small point and a matter of opinion, but 
is worth mention. The involute is a simple and graceful single curve, while 
the epicycloid is a double and not mechanically a neat curve, and, as gener- 
ally drawn, has a decided bulge or even a plain corner where the two halves 
join at the pitch line. 

In General. As the involute has the advantage of the epicycloid, in 
Ijine actual cases out of ten, with respect to adjustibility in position, in 
uniformity of wear and action, in loss of power and change of shape by 
friction, in strength, and in appeaiance, and is but a shade, if any, inferior 
with regard to the thrust on the bearings, it may be, and should be accorded 
first place for any and every practical purpose. The writer can imagine no 
possible case, unless it be in connection with a pinion of very few teeth, 
where the epicycloid would have either a theoretical or a practical advan- 
tage over the involute. 



ODONTOGRAPH TABLE, 
INVOLUTE TEETH. 

Corrected for Interference, 
Interchangeable Set. 





DIVIDE 


BY THE 


MULTIPLY BY THE 




DIAMETRAL 


CIRCULAR 


TEETH, 


PITCH. 


PITCH. 


Face 


Flank 


Face 


Flank 




Radius. 


Radius. 


Radius. 


Radius. 


12 


2.70 


.83 


.86 


.27 


13 


2.87 


.93 


.91 


.30 


14 


3.00 


1.02 


.95 


.33 


15 


3.15 


1.12 


1.00 


.36 


16 


3.29 


1.22 


1.05 


.40 


17 


3.45 


1.31 


1.09 


.43 


18 


3.59 


1.41 


1.14 


.46 


19 


3.71 


1.53 


1.18 


.50 


20 


3.86 


1.62 


1.22 


.53 


21 


4.00 


1.73 


1.27 


.57 


22 


4.14 


1.83 


1.32 


.60 


23 


4.27 


1.94 


1.36 


.63 


25 


4.56 


2.15 


1.45 


.70 


28 


4.82 


2.37 


1.54 


.77 


31 


5.23 


2.69 


1.67 


.88 


34 


5 77 


3.13 


1.84 


1.00 


38 


6.30 


3.58 


2.01 


1.16 


44 


7.08 


4.27 


2.26 


1.38 


52 


8.13 


5.20 


2.59 


1.70 


64 


9.68 


6.64 


3.09 


2.18 


83 


12.11 


8.93 


3.87 


2.90 


115 


16.18 


12.80 


5.16 


4.15 


200 


25.86 


22.30 


8.26 


7.30 



For intermediate teeth use proportionally intermediate values when great 
accuracy is desired, but for drafting purposes use the nearest value, thus : — 
35 is at one-quarter of the distance from 34 to 38, and the proper values for 
accurate work are : face radius, 5.90 inches, and flank radius 3.24 inches. 

The table is not carried beyond 200 teeth, as the higher numbers are rarely 
used and the radii are then very great. For drafting purposes use values for 
200 teeth for all higher numbers. 

The base distance, the distance from pitch line to base line, is always one- 
sixtieth of the pitch diameter. 

SPECIAL PROCESS FOR RACK TEETH. 

See the cut on the opposite page. 

The flank of the tooth and one-half of the face is a straight line at an angle 
of 75 degrees, five-sixths of a right angle, with the pitch line. 

Draw the outer half of the face of the tooth, one-quarter of its whole length, 
as a circular arc from a center on the pitch line and with a radius of 
2.10 inches divided bv the diametral pitch. 
.67 inches multiplied by the circular pitch. 

The point must be rounded over in this way to avoid interference, if the- 
rack is to mesh with any pinion having less than 28 teeth. 



A PRACTICAL EXAMPLE. 




INVOLUTE TEETH. 



INTERCHANGEABLE SET. 



Example. — A rack, and a pinion of twelve teeth, of two diametral pitch. 

Pinion. — From the tables we have, after dividing by 2, the face radius 1.35 
inches, flank radius .42 inches, and clearance .06 inches, The pitch diameter 
is 6 inches, and the addendum is .5 inches. The base distance, one-sixtieth 
of the pitch diameter, is .10 inches. 

Draw the pitch line and divide it for the pitch points, allowing for backlash 
if required. Lay off the addendum and the clearance, and draw the adden- 
dum line, root line, and clearance line. 

Draw the base line .10 inches inside the pitch line. 

With the face radius, 1.35 inches, and from centers d on the base line, draw 
all the face curves from addendum line to pitch line. With the flank radius, 
.42 inches, and from centers h on the base line, draw all the flanks from the 
pitch line to the base line. 

The flanks inside the base line are stra'ight radial lines. 

For fifty or more teeth draw the flank curve from pitch line to root line. 

Rack. — Draw by the special rule, the radius for the point being 1.05 inches. 

Note. — The dotted lines on the pinion teeth show the work of the common 
rule for involute teeth, as explained on page 18 and given by most of the 
" gear charts " and works on practical mechanism. The same rule draws the 
rack tooth with a point that is not rounded. The " old rule " is as worthless 
as it is simple. 





/y ^^ 



r 



o v- 



\P 




BEVEL GEARS. 

In layinpj out the teeth of a bevel gear but one new point needs to be con- 
sidered. The working pitch diameter a b c is not to be used, but the teeth 
are to be drawn on the conical pitch diameter ad c, developed or rolled out 
as in fig. 25. 

The conical diameter a d c may be found from a drawing, or if the gears 
are of some common proportion, from the following table by multiplying 
the true pitch diameters by the tabular numbers given for that proportion, 

TABLE OF CONICAL PITCH DIAMETERS 
OF BEVEL GEARS. 



Proportion. 


Larger Gear. 


Smaller Gear. 


1 tol 


1.41 


1.41 


2 " 1 


2.24 


1.12 


3 " 2 


1.80 


1.20 


3 " 1 


3.16 


1.05 


4 " 3 


1.67 


1.25 


4 " 1 


4.12 


1.03 


5 " 4 


1.60 


1.28 


5 " 3 


1.94 


1.17 


5 " 2 


2.69 


1.08 


5 « 1 


5.10 


1.02 


6 " 5 


1.56 


1.30 


6 " 1 


6.08 


1.01 


7 " 1 


7.07 


1.01 


8 " 1 


8.06 


1.01 


9 " 1 


9.06 


1.01 


10 '' 1 


10.05 


1.01 



Examples.— A miter gear, proportion 1 to 1, of 4 pitch, 6'' diameter, and 
24 teeth, has a conical diameter of 6" x 1.41 = 8.46'', and there are 24 x 1.41 
= 33.8 teeth on the full circle of the developed cone. 

A pair of bevel gears of 3 to 1 proportion, 48" and 16'' diameters, 36 and 12 
teeth, have conical diameters 48" x 3.16 = 151.68", and 16" x 1.05= 16.80", 
and there are 36 x 3.16 = 113.76, and 12 x 1.05 = 12.60 teeth on the full cir- 
cles of the developed cones. 



^^^^ 



wm "'^^ij^ 4. ._'^''^V 




INTERNAL GEARS. 

The iuterD'i gear, sometimes called the "annular" gear, is drawn by the 
rules for spur gears, the teeth of a spur gear being the spaces between the 
teeth of an internal gear of the same pitch diameter, with the backlash and 
clearance reversed in position. 

Involute teeth should end at the base line, the radial part of the flank 
being omitted, or well rounded over if it is desirable to preserve the appear- 
ance of the full tooth. 

Internal teeth will interfere, even if properly drawn, unless the gear is 
considerably larger than the pinion running in it. If drawn for the common 
twelve to rack interchangeable set, there should be at least twelve more 
teeth in the gear than in the pinion, and if the difference is less, the teeth 
must be " doctored " or rounded over until they will pass, and there must 
be a difference of two teeth in any case. 

Involute teeth have a decided advantage over epicycloidal teeth for inter- 
nal gearing, their action being much more direct, with less sliding Jitid 
friction. 



STRENOTH ANDIHOESE-POWEE OF OAST GEAES. 



There are about as many different rules for this purpose, and contradictory re- 
sults, as there are writers upon the subject. I have preferred not to discuss the 
theory, but to adopt without question the method given by Thomas Box in his Prac- 
tical Treatise on Mill Gearing, because that engineer has most carefully considered 
the practical points in view, and because his formulae agree almost exactly with a 
great many cases in actual practice. 



STRENGTH OF A TOOTH. —For worm gears, crane gears, and slow-moving 
gears in general, we have to consider only the dead weight that the tooth can lift 
with safety. 

If we allow the iron to be subjected to but one tenth of its breaking strain, we 
can use the formula: — 

W = 350 c f , 
in which W is the dead weight to be lifted, c is the circular pitch, and f the face, 
both in inches. 

For the wooden cogs of mortise wheels, use 120 instead of 350 as a factor in the 
fonnula. 

When the pinion is large enough to insure that two teeth shall always be in fair 
contact, the load, as found by this rule, may be doubled. 

Example. — A cast-iron gear of 3" circular pitch and 6" face will lift 
W = 350X3X6 = 6300 lbs. 



HORSE-POWER OF A GEAR. — For very low speeds we can use the 

formula, 

HP for low speed = .0037 d n c f , 
in which d is the pitch diameter, c the circular pitch, and f the face, all in inches, 
and n is the number of revolutions per minute. 

Example. — The horse-power of a gear of three feet diameter, three inch 
pitch, and ten inch face, at eight revolutions per minute, is, 
HP = .0037 X 36 X 8 X 3 X 10= 32. 



For ordinary or high speeds, where impact has to be considered, it is found that 
the above formula gives too high results, and we must use the formula, 
HP at ordinary speeds = .012 c^ f -\/du. 

Example. - A gear of three feet diameter, three inch pitch and ten inch face, 
at one hundred revolutions per minute, will carry but 

HP = .012 X 9 X 10 X v'lOO X 36 = 65 horse-power, 
instead of the 400 horse-power found by the rule for low speeds. 



At ordinary or high speeds a wooden cog, on account of its elasticity, will carry 
as much as or more power than a cast-iron tooth, and we can use .014 instead of .012 
in the formula. 

When in doubt as to whether a given speed is to be considered high or low, com- 
pute the horse-power by both formulae, and use the smallest result. 



For bevel gears the same rules will apply, if we use the pitch diameter and the 
pitch at the center of the face. 

Some rules in use take no account of the face of the gear, but assume that the 
tooth should be able to bear the whole strain upon one corner. 

A tooth that does not bear substantially along its whole face, at several points at 
least, is a very poor piece of work, and it would be better to straighten the tooth 
than to force the rule to follow it. 



HORSE POWER OF CUT GEARS. 

The rules giveu above for the horse power of gears apply only to gears 
with rough cast teeth ; and in applying them we must consider the speed of 
the gear as well as its real strength. 

One of the chief sources of weakness in a cast gear, is that the continual 
pounding of the teeth on each other crystalizes the metal so that its strength 
is gone long before it is worn out. 

There are no recorded tests on the horse power of cut gears, but it is gen- 
erally agreed among those not personallj^ interested in the sale of cast gear- 
ing, that a cut gear is much more durable, and that it will carry more power 
than a cast gear, with the same factor of safety. 

In the absence of experimental data, we can only proceed by judgment and 
inference. It is well settled that the continual pounding of cast gearing is a 
source of weakness that must be allowed for, and it may be assumed that 
that source is avoided in the use of cut gears having a smooth and even 
action. 

Until practical tests have been made we can consider that the rule that 
applies to cast gears for slow speeds where impact need not be considered, 
can safely be applied at higher speeds to cut gears where there is no impact 
to be allowed for ; and we have the formula : — 

Horse power of cut gears at ordinary speeds = .0037 dncf. 

Applying this formula to the case of a gear of 36 inches diameter and 3 
inch circular pitch, at 100 revolutions per minute, it is found that the cut 
gear will safely carry six times the power that can be trusted to the cast gear. 

But it must be admitted that all that is known concerning the real horse 
power of a cut gear is a matter of inference, and it is to be hoped that the 
growing use of cut gearing for conveying heavy powers will furnish data of 
a more practical and trustworthy nature. Until such data is at hand it may 
safely be assumed that a cut gear has from two to three times the carrying 
power of a rough cast gear of the same size. 



CONFUSION OF RULES. 

The disagreement of standard authorities and the thorough confusion of 
rules on this subject, is well shown in an interesting paper by J. H. Cooper, 
in the Journal of the Franklin Institute for July, 1879, in which that engineer 
has industriously collected twenty-four formulas from Tredgold, Buchanan, 
Fairbairn, Box, Molesworth, Haswell, Nystrom and others, and applied them 
to the practical case of a gear of 60 inches diameter, 7k inches face, and 3 
inches pitch, at 60 revolutions per minute. Cooper found twenty-two differ- 
ent results for this one example, as follows: — 46.31, 47.06,50.27, 53.18,56.09, 
56.55, 63.62, 66.17, 66.27, 67.96, 68.56, 73.49, 80.78, 84.37, 86.75, 86.80, 86.96, 
138.23, 147.27, 163.00, 294.53, and 295.59. Here is variety to suit all tastes, 
and if a gear is not strong enough for a given purpose according to Fair- 
bairn, it will certainly fill the bill according to Haswell. Diligent enquiry 
by myself among the cast gear makers of the United States gave the 
same result as to variety and confusion. I could get little but opinions 
that were not founded on experiment, and the opinions were of the most in- 
definite and unsatisfactory character. 

All cast gear makers are agreed that a cast gear is more durable than a cut 
gear, and all cut gear makers are equally certain that a cut gear is more du- 
rable than a cast gear. 

The stock argument of the makers of cast gearing is that the one-hundredth 
of an inch thickness of hard scale on a cast tooth makes it more durable 
than a cut gear from which the scale has been removed. But, from that 
point of view, they find it very hard to explain why a mortise gear, with 
soft hickory cogs, is quite as durable as a cast gear with hard teeth. 



CHART AND TABLES 



FOR 



BEVEL GEARS. 



A NEW, SIMPLE, AND ACCUEATE 3IETH0D EOR FINDING THE ANGLES 
AND DIAMETERS OF ANY PAIR OF BEVEL GEARS BY SIMPLE 
CALCULATION, AND WITHOUT DRAFTING INSTRU- 
MENTS OR SPECIAL TOOLS. 



J^ot one machinist in a dozen will admit that he does not knosv how to 
properly shape a bevel gear blank, but when put to the test, not one in a 
score can do it well without an amoimt of fussing with drafting instru- 
ments, and a deal of studying and figuring that looks ridiculous to one who 
has studied the subject and knows how simple it really is when it is once 
thoroughly understood. 

The average bevel gear blank can be relied upon to be wrong in i ts face 
angle, or its outside diameter, or both, even when it has been shaped by a 
competent and intelligent general workman, and the simple explanation is 
that the only reliance is generally a hurried and poorly made drawing from 
which the angles and diameters must be found by measurement, and used 
with many chances of error. 

This method proceeds by simple calculation, avoiding the use of drafting 
instruments, and it will be found to be not only much more accurate, but at 
tlie same time much easier and quicker than any other method. The work- 
man wlio can remember the numbers 1.41 and 81 and the angle 45° needs 
no further assistance on miter gears, and on other proportions needs the 
table only to supply equally simple data, while two to live minutes is suffi- 
cient time for any set of calculations after the method has been learned. 

This matter is of more importance than is generally supposed, for bevel 
gears unlike spur gears must be exactly correct in diameters and angles, 
or no amount of perfection in the cutter or care in the cutting will prevent 
a botch. 

To be learned this Chart must be studied. If it is not worth 
while to give it two or three hours of careful attention it is not worth while 
to keep it at all. It is simple and easy to learn but it cannot be taken in at 
a glance, or comprehended in ten minutes. 




EXPLANATION OF THE METHOD. 

The measures that must be 
<?iven in advance are the pitch 
diameters AB and AD, and 
the numbers of teeth or the 
I diametral pitch, and the meas- 
ures that must be determined 
before the blank can be shap- 
ed are, 

1st, the outside diameters 
m n and p q, each equal to the 

*^^ |£^»^/{?^«'2«:_^ff pitch diameter plus a small 

\S I / increment. 

2d, the center angles A C M 
and A C X. 

3d, the face angles cCM 
and bCI^, each equal to the 
center angle plus a small in- 
crement. 

4th, the cutting angles a C M 
and d C X, each equal to the 
center angle minus a small 
increment. 
Of course all these can be 
'*!.'.-. ^*/..— found by first making a draw- 

ing and then measuring the diameters and angles, but, although the 
process is simple enough, and perhaps preferable in some cases in the hands 
of a draftsman "who has the proper tools to use and the skill and knowledge 
to use them properly; still it is not well adapted to ready and general use 
in the shop. 

Unless made with good instruments that are handled with great care, a 
drawing is not accurate enough for the purpose, and its results, particularly 
as to angles, are not apt to be well carried out on the work. It is easy 
enough to find the proper face angle by a drawing, but not as easy to 
measure that angle and transfer it to the iron blank. 

This method is entirely one of simple calculation, with no instrument but 
the pen, and no tools to apply its results in the shop but the ordinary scratch- 
ing dividers and the scale. It is not only more accurate, but, after a few 
hours work with the table on various examples, it will be easier to work 
and quicker than the graphical method. 

PROCESS IN DETAIL. 

First find the proportion of the gears by dividing the diameter of the 
larger gear by that of the smaller, and then use the values in the table 
opposite that proportion. 

The diameter increments are found by dividing the tabular increments by 
the pitch, and the outside diameters are found by adding the increments to 
the pitch diameters. 

The angle increment is found by dividing the tabular increment by the 
number of teeth in the larger of the two gears. The face angle is the 
center angle plus the angle increment, and the cutting angle is the center 
angle minus one and one-sixth of the increment. 

Example. — Given pitch diameters 6" and 4", pitch 8, teeth 48 and 32. 
The proportion is 6 to 4, or 3 to 2, or 1.5 to 1, and the table gives at 1.5 the 
center angles 56.3° and 33.7°, and the angle increments ff =: 2°, and 11 of 2° 
= 2.3°, from which we find the face angles 5G.3°-f 2° = 58.3°, and 33.7^ + 2° 
= 35.7°, and the cutting angles 56.3° — 2.3° = 54°, and 33.7° — 2.3° = 31.4°. 

1.11 1.66 

The diameter increments are = .14, and = .21, and from these we 

8 8 

find the outside diameters 6. -f .14 = 6.14'', and 4. -f- .21 = 4.21''. 

When the proportion falls between two tabular proportions we must use tab- 
ular angles and increments that, are proportionally between the tabular values. 

All diameters should be figured to the nearest hundredth of an inch, and 
all angles to the nearest tenth of a degree. 



PRACTICAL APPLICATION IN THE SHOP. 

The best tool for shaping a bevel gear is a compound and graduated 
slide rest which can be set directly to the angles, but as such an appliance 
is seldom to be found, and then seldom in good order, the work must gen- 
erally be done by the use of angle templets. 

To use the angle templet, lay one edge against a straight shaft or mandrel 
held between centers, or against the tail spindle, and adjust a side tool or 

scraper to the other edge. If 
the back of the templet be cut 
square with one edge it can be 
used by placing it against the 
face plate. 

Be careful with the back 
angle, for the back of the tooth 
is not square with its face, and 
if it is turned square or in any 
way out of truth, the corners 
of the teeth will not fit true in 
the spaces of the mate gear, 
and the appearance of the job 
will be spoiled. The true back 
angle of each gear is the cen- 
ter angle of the other gear of 
the pair. 

SHAFTS NOT AT RIGHT ANGLES. 

When, as sometimes happens, the shafts are not at right angles, a simple 
preliminary drawing must be m;ide. Lay oif the shaft angle by means of 
the table of chords, draw two lines parallel to the shafts at the distances of 
the half pitch diameters, and from their intersection draw a conical line to 
the center. 

Measure the center angle between either shaft and the conical line, and the 
proper increments will be found in the table at that center angle. If the 
center angle is less than 45°, the tabular angle increment must be divided 
by the number of teeth in the gear, as usual, and then divided by the 
tabular proportion. 

Example. — It is found that the center angle of a gear of 8 pitch and 40 
teeth is 35°, and this in the table gives the 




94 



anale increment 



1.63 
and the diameter increment = .20 



1.43 X 40 



Proceed separately with the 'Other gear of the pair by measuring and using 
its center angle. 

ERRORS IN DIAMETER. 

It will often happen that the outside diameter vtdll be turned too small, or 
that a casting will not quite turn to the desired size. In this case the 
diameter should be left as large as possible, and then the other, or mate 
gear, should be turned under size, to keep the correct proportion between the 
pitch diameters. 

For example, if the smaller gear of a pair that are proportioned two to 
one is f oiind to be gV inch under size, the larger gear must be turned twice 
as much, or -^ inch under size. 

Great care must be taken to have the smaller gear of an unequal pair very 
near to size, for any inaccuracy can be balanced only by a proportionally 
larger inaccuracy of the mate gear. If the smaller gear of a pair that are 
proportioned six to one is as much as -^^ of an inch under size, the larger 
gear must be turned g^, or y\ inch under size, and this is enough to change 
the number of teeth the proportion and the face, and spoil the work. The 
only remedy in such a case is to cut shallow teeth on the pinion, so that its 
pitch diameter is unchanged. 



ANGLE TEMPLETS. 




To make an angle templet by the use of 
the table of chords, draw an arc ad on 
paper or sheet metal with the dividers set 
to six inches. Then set the dividers to the 
chord of the angle and lay it oft' on the arc, 
as at b c. Cut to the lines b o and c o. 

Similarly, an angle drawn on paper can 
be measured by drawing an arc across it 
at a radius of 6", measuring the chord, 
and comparing with the table. 



TABLE OF CHORDS OF ANGLES, 

AT RADIUS OF SIX INCHES. 



Degrees 


Chord. 


Tenths. 


Degrees 


Chord. 


Tenths. 


Degrees 


Chord. 


Tenths. 


1 


.10 




31 


3.20 




61 


6.10 




2 


.20 




32 


3.31 




62 


6.19 




3 


.31 




33 


3.41 




63 


6.28 




4 


.42 




34 


3.51 




64 


6.36 




5 


.52 




35 


3.61 




85 


6.45 




6 


.62 


36 


3.71 


66 


6.54 


7 


.73 




37 


3.81 




67 


6.62 




8 


.84 




38 


3.91 




68 


6.71 




9 


.94 




39 


4.01 




69 


6.80 




10 


1.04 




40 


4.10 




70 


6.89 




11 


1.15 


41 


4.20 


71 


6.97 


12 


1.26 




42 


4.30 




72 


7.06 




13 


1.36 


.1— .01 
.3— .02 
.2— .03 
.4— .04 


43 


4.40 


.1— .01 
.2— .02 
.3— .03 
.4— .04 


73 


7.14 


.1— .01 
.2— .02 
.3— .02 
.4— .03 


14 
15 


1.46 
1.57 


44 
45 


4.50 
4.60 


74 
75 


7.22 
7.31 


16 


1.67 


46 


4.69 


76 


7.39 


17 


1.77 


.5— .05 


47 


4.79 


.5— .05 


77 


7.47 


.5— .04 


18 


1.87 


.6— .06 


48 


4.88 


.6— .05 


78 


7.55 


.6— .05 


19 


1.98 


.7— .07 


49 


4.98 


.7— .06 


79 


7.63 


.7— .06 


20 


2.08 


.8— .08 
.9— .09 


50 


5.08 


.8— .07 
.9— .08 


80 


7.71 


.8— .06 
.9— .07 


21 


2.18 


51 


5.17 


81 


7.79 


22 


2.29 




52 


5.26 




82 


7.87 




23 


2.39 




53 


5.35 




83 


7.95 




24 


2.49 




54 


5.45 




84 


8.03 




25 


2.59 




55 


5.54 




85 


8.11 




26 


2.70 


56 


5.63 


86 


8.18 


27 


2.80 




57 


5.72 




87 


8.26 




28 


2.90 




58 


5.82 




88 


8.34 




29 


3.00 




59 


5.91 




89 


8.41 




30 


3.10 




60 


6.00 




90 


8.48 





The table gives the length of the chord at six inches from center, for any 
degi-ee. For tenths of a degree, add the value in the small table of the 
same coliunn. 



TABLE OF 

INCREMENTS AND ANGLES 

FOR 

BEVEL GEARS. 



Proportion. 


Diameter Increment. 
Divide by pitch. 

Larger Smaller 
Gear. Gear. 


Angle 
Increment 

Divide 
by number 
of teeth in 
larger gear 


Center Angles. 

Larger Smaller 
Gear. Gear. 


1.00 
1.05 
1.10 
1.11 
1.13 


1-1 

10-9 

9-8 ■ 


1.41 

1.37 
1.35 
1.34 
1.33 


L41 
1.42 
1.44 
i.46 
1.47 


81 

84 
86 
87 
87 


45.0 
46.4 

47.7 
48.0 
48.5 


45.0 
43.6 
42.3 
42.0 
41.5 


1.14 
1.15 
1.17 
1.20 
1.25 


8-7 

7-6 
6-5 
5-4 


1.32 
1.31 
1.30 

1.28 
1.25 


1.49 
1.50 
1.52 
1.54 
1.56 
1.58 
1.59 
1.60 
1.61 
1.62 


88 
89 
89 
90 
91 


48.7 
49.0 
49.5 
50.2 
51.3 
52.2 
52.4 
53.1 
53.5 
54.5 


41.3 
41.0 
40.5 

39.8 

38.7 


1.29 
1.30 
1.33 
1.35 
1.40 


9-7 
4-3 

7-5 


1.24 
1.22 
1.20 
1.18 
L16 


91 
92 
93 
93 
94 


37.8 
37.6 
36.9 
36.5 
35.5 


1.43 
1.45 
1.50 
1.55 


10-7 
3-2 


1.15 
1.13 
Lll 
1.09 


1.63 
1.65 
1.66 
1.67 


94 
95 
95 
96 


55.0 
55.4 
56.3 

57.2 


35.0 
34.6 
33.7 

32.8 


1.60 
1.65 
1.67 
1.70 
1.75 


8-5 
5-3- 

7-4 


1.07 
1.05 
1.03 
1.01 
.99 


1.68 
1.70 
1.72 
1.73 
1.74 


96 

97 

98 

99 

100 

101 

101 

101 

102 

102 


58.0 
58.8 
59.1 
59.5 
60.3 
61.0 
61.6 
62.2 
62.8 
63.5 


32.0 

31.2 
30.9 
30.5 

29.7 


1.80 
1.85 
1.90 
1.95 
2.00 


9-5 
2-1 


.97 
.95 
.93 
.91 

.89 


1.75 
1.76 
1.77 
1.78 
1.79 


29.0 

28.4 
27.8 
27.2 
26.5 


2.10 

2.20 
2.25 
2.30 
2.33 


9-4 
7-3 


.87 
.84 
.82 
.80 
.78 


1.80 
1.81 
1.82 
1.83 
1.84 
1.85 
1.86 
1.86 
1.87 
1.87 


103 
103 
104 
104 
105 
i05 
106 
106 
107 
107 


64.6 
65.5 
66.1 
66.5 
66.8 
67.4 
68.2 
68.9 
69.5 
69.7 


25.4 
24.5 
23.9 
23.5 
23.2 


2.40 
2.50 
2.60 
2.67 
2.70 


5-2 
8-3 


.76 
.75 
.73 
.71 
.69 


22.6 
21.8 
21.1 
20.5 
20.3 


2.80 
2.90 
3.00 
3.20 
3.33 


3-1 
10-3 

7-2 

4-1 


.67 
.65 
.63 
.60 
.58 


1.88 
1.89 
1.91 
1.92 
1.92 
1.92 
1.93 
1.93 
1.94 
1.94 


108 
108 
109 
109 
109 


70.3 
71.0 
71.6 

72.7 
73.3 


19.7 
19.0 
18.4 
17.3 
16.7 


3.40 

3.50 

1 3.60 

1 3.80 

1 4.00 


.56 
.54 

.52 
.50 
.49 


110 
110 
110 
111 
111 


73.6 
74.1 
74.5 

75.2 
76.0- 


16.4 
15.9 
15.5 
14.8 
14.0 



APPENDIX 



A FEW PAPERS ON THE 

TEETH OF GEARS, 

REPRINTED FROM THE 

"AMERICAN MACHINIST" 

AND THE 

"JOURNAL OF THE FRANKLIN INSTITUTE." 



These papers are not of a popular or so-called practical character, but they may 

be of interest to the student. There are a nunnber of articles in 

the same periodicals, not reprinted here. 



THE NORMAL THEORY 



OF THE 



GEAR TOOTH CURVE. 



The usual method of presenting the gear 
tooth curve for the examination of the student, 
is to devote almost exclusive attention to the 
minute details of the cycloidal system, to 
hurry over the involute system, and to get the 
general tooth curve, without regard to any 
special form, into the smallest possible com- 
pass. 

The result, to the student, is a more or 
less intimate knowledge of the cycloid, with 
a fixed idea that it is the only tooth curve 
worth his serious attention, a smattering, 
generally wrong at that, with regard to the 
involute, and little or no acquaintance with 
the curve in its general and most interesting 
condition. 

The subject should be begun at the begin- 
ning, and the beginner should learn what an 
" odontoid," or pure tooth curve is, what it 
does, and how it does it, before he is plied 
with epicycloids, logarithmic spirals, and the 
less important details of its special forms. 

When properly treated, the gear tooth curve 
is not difficult to explain or understand, and 
is one of the most interesting and im- 
portant applications of mathematics to prac- 
tical mechanics. 




THE NOEMAL. 

A normal to any curve is a straight line 
i\r, Fig. 1, which is at right angles with it at 
the point of intersection. 



Fig,2 




Curve andnormaljs 



Moiling ivJieels 



KOLLING WHEELS. 

If two friction wheels. Fig. 2, roll on each 
other, their pitch lines touching at the pitch 
point 0, they are, for the instant, revolving 
about centers A and B, which are in ^ OB, 
the common normal or line of centers of the 
two curves at their common point. 

ENVELOPING TEETH. 

If teeth of any arbitrary shape, Fig. 3, are 
fastened to one rotating wheel A, they can 



THE NORMAL THEOEY OF THE GEAR TOOTH CURVE. 



be made to form teeth on a blank disk fast- gate and others are not, the property is evi- 
ened to the other wheel B, which are the dently due to some peculiarity of their shape. 



bounding curves, or " envelopes' 
different positions. 



Fig.3 




of all their Conjugate enveloping teeth will form the 
originating teeth by the same process, re- 
versed, on a new wheel blank like the original, 
but if not conjugate they will not always re- 
produce the originals. 



FigA 



UnveZopiiig teeth 



These enveloping teeth can be formed by 
scribing lines about the originating teeth at 
short intervals of their motion, and then cut- 
ting out all the lines ; or, if they are cutting 
tools, which reciprocate vertically, see Fig. 
13, they will cut them out as the friction 
wheels roll together. 

CONJUGATE TEETH. 

If the originating teeth be passed again 
through the enveloping teeth they have 
formed, the friction wheels rolling together as 
at first, they will not, of course, interfere with 
them or cut them again. With originating 
teeth of certain shapes they will entirely 
separate at times, and touch each other at 
other times, while teeth of certain other 
shapes will have the peculiar property that 
they will continually touch each other, and 
not separate at all from the first touch to the 
last. 

The latter property is evidently of the 
greatest mechanical value, for teeth that have 
it can be used to forcibly transmit a uniform 
motion from one rotating wheel to another, 
without the feeble and uncertain assistance of 
the friction wheels. 

Such teeth are said to be " conjugate" to 
each other, and their curves are called 
** odontoids," and, as some forms are conju- 




N'ormal intersection 

THE LAW OF NORMAL INTEESECTION. 

If the two odontoids, iV' and M, Fig. 4, are 
conjugate they are always in driving contact. 
They must always be tangent and not inter- 
sect at some one point P, and they must have 
a common normal, P, at that point, which 
will intersect the line of centers, A B, at 
some point 0. 

The two wheels have a common velocity, 
P Q, of their common point of contact, P, 
along the common normal P 0, and any com- 
mon point 0, on that normal, has the same 
common velocity in the same direction. 

The two wheels can have a common velocity 
on the line of centers only at their common 
pitch point; therefore, the point is that 
pitch point, and the first and most important 
law of the action of the odontoid is : 

TJie normal to the point of contact of an 
odontoid with any other odontoid^ always passes 
through the pitch point. 



THE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



THE ODONTOID. for Want of a more expressive term, may be 

This law determines the general nature of called ** consecutive," and our general defini- 

the odontoid. tion is : 

If KP'S, Fig. 5, is an odontoid the points -^ny curve Mmng consecutive normals to the 

K 0' 0\ etc., of the pitch line will pass the P^t<^^ ^^'^^ **« a practicable odontoid. 

pitch point consecutively, each one in its 

order, without a break, or a return in the 

order from the first point K to the last one 

used. 



Fig.6 





The odontoid 



Semicircular teeth 



For an example of a curve that is not an 
odontoid, although it is often treated as such, 
the semicircular teeth of the rack of Fig. 6 
form enveloping teeth on the pinion which 
are circles of the same radius. All the nor- 
mals to the circle intersect the pitch line at 
When any point, P,' of the odontoid be- the center/, and the tooth will not touch the 
comes the point of contact P, the normal space until the centers / and g come together 
from it must pass through the pitch point 0, at 0. Then all the normals satisfy the law. 
and, as contact must be continuous, there and the tooth fits and coincides with the 
must be a normal to the odontoid from every space. 

point of the pitch line. The absence of normal intersections on any 

It is also a feature of any practicable odon- part of the pitch line shows that the teeth will 
toid that its point of contact continually shifts separate when that part is passing the pitch 
on it, a new point on the curve coming into point, and the junction of two or more normal 
contact as each point on the pitch line comes intersections will show that the teeth will co- 
to the pitch point, without a break or a return incide at that place. 

in the consecutive order. It has been stated that any assumed and 

When there are many normals from nearly the arbitrary rack tooth, within the limitation 



same point on the curve, that point is in ex- 
cessive use, and such a curve, although possi- 
ble, is not useful. When the normals cross 



that it is " bounded by four similar and equal 
lines in alternate reversion, * * * -^n 
form an interchangeable set."* The semi- 



each other there will be a cusp formed on the circular rack tooth is clearly within the given 
odontoid, and it is impracticable. limitation, but it is not an odontoid, and will 

As both ends of each normal must come not form an interchangeable set. The conju- 
into position in order, one after the other, gate tooth curve is subject to a law that is very 
they must be arranged as in Fig. 5, elastic, but by no means indefinite, and which 
one after the other, without a crossing, and is seldom clearly given and often is given 
without a blank interval. This arrangement, wrong by writers on this subject. 

♦MacCord's Kinematics, section 283, and again at section 408. No conic section is an odontoid unless 
the focus is inside the pitch luie, but thej' will all meet Prof. ^VlacCord's reciuirement. 



THE NORMAL THEORY OP THE GEAR TOOTH CURVE. 



Fig. 7 




FACES AND FLANKS. 

The odontoid is on one side only of its pitch 
line and will be conjugate only to a similar 
odontoid on the same side of any companion 
pitch line, all being faces on one gear and 
flanks on the other. 

Face gears and flank gears will work to- 
gether but two face gears or two flank gears 
will not. To make a completely interchange- 
able set it is therefore necessary to provide 
each gear with both faces and flanks, all being 
similar odontoids, as in Fig. 8. 



Similar odontoids 



INTERCHANGEABIiE ODONTOIDS. 

If any number of odontoids, Fig. 7, are 
formed on the same side of a set of pitch 
lines that will all roll together at the pitch 
point 0, and ail the odontoids conform to the 
requirement that the normal arcs IC, nor- 
mals P' K and normal angles P' K S shall 
all be the same, they maybe called " similar " 
odontoids. 

As any two of these pitch lines with their 
odontoids roll together, the points K will 
pass over the same arcs and come together to 
the pitch point at the same time, and as 
the normals and normal angles are equal, the 
two points P' will come together at P. The 
two odontoids are therefore always in driving 
contact, without regard to the curvature of 
the pitch lines, and the general law of inter- 
changeability is: 

A II similar odontoids will work in interchange- 
able contact with each other. 



mg.8 




Fig,9 




Unrevevsihle teeth 



EEVEKSIBLE GEAES. 

If the two sides a b and cd of the tooth of 
Fig. 9 are not similar odontoids, the teeth 
may still belong to an interchangeable set, for 
the similar sides will always come together, no 




Comjylefe teeth 



Whole teeth 

matter how the gears are interchanged. But if 
one gear of such a set is turned over, re- 
versed face for face, the unlike odontoids a b 
and c d will come together, and it is neces- 



THE NORMAL THEORY OF THE GRAB TOOTH CURVE. 



sary, to have the set reversible, that all four 
odontoids al) c and d, bounding the tooth, 
shall be similar, 

TRUNCATED TEETH. 

When the point of contact P of the whole 
tooth, Fig. 10, is near the pitch line, its 
action on the mating tooth is nearly a direct 
push, but it becomes more and more oblique, 
with a wedging or crowding, as well as a 
pushing action, until, at the apex q there is 
no driving action at all, and the driven gear 
will stop unless both gears are so large that 
the next tooth is then in position, as in 
Pig. 8. 





Truncated teeth 

To avoid this oblique action and at the same 
iime allow the use of gears of few teeth, it is 
customary to truncate or cut off the apex of 
the curve, as shown by Fig. 11, by a line, 
called the addendum line, at an arbitrary dis- 
tance from the pitch line. 

The sides of the teeth are then brought as 
near together as is consistent with the re- 
quired strength and we have the familiar tooth 
in universal use. 

rOEMS OF TOOTH-CURVES. 

The face curve, commonly the external 
curve, is a lobe m^ Fig. 12, which gener- 
ally returns to the pitch line, but the flank 
or internal curve may take a variety of 
shapes, generally a loop d or h, but 
sometimes a straight diameter 7c, a point at 
0, a loop Of,& cusp a n, or a double cusp 
O a' b c. 

"When the flank is undercurved as a.t h, & 



Tooth curves 



weak tooth is formed that should be avoided, 
and when it is nearly a point at 0, it is sub- 
ject to such excessive wear as to be impracti- 
cable. 

When a cusp, a n, ov a' b cis foiiaed, 
the action is mathematically perfect at all 
points, but practically is limited to the first 
branch from to a. The contact changes, at 
the cusp, from one side of the line to the 
other, and is therefore impracticable with 
real teeth. The cusp always sets a limit to 
the addendum of the tooth that is working 
with it, for that tooth, as it continues in 
mathematical contact with the second branch, 
will interfere with and cut away the first 
branch. 



Fig. 13 




The con jug at or 



THE CONJUGATOR. 

We have seen, Fig. 3, that any odontoid 
will form or " develop," an enveloping curve, 



THE NORMAIi THEORY OF THE GEAR TOOTH CURVE. 



that is also an odontoid and coDJugate to it. 
The same process provides a simple and exact 
method for forming templets and cutter- 
shapers, in the application of the theory to 
practical purposes. 

Theconjugator, Fig. 13, is here the connect- 
ing link between theory and practice, for if 
the gear-cutter, or templet, must be shaped 
by hand and eye processes, theoretical pre- 
cision would be lost, and the perfection of the 
finished product would depend, as usual, 
more on personal skill than on original prin- 
ciples. 

A rack tooth is first formed on a steel cut- 
ting tool A, which is fastened to a slide F 
that reciprocates vertically on a stand, B, on 
a plane table, H. A straight-edge, G, is 
fastened to the table in the position of the 
pitch line of the rack tooth, and an arc, K, 
representing the pitch line of the tooth to be 
formed, is rolled against it. A steel band, 
D B, keeps the arc firmly in position on the 
straight edge, weights GOG keep it in 
position on the table, and a screw, M, serves 
to slowly roll it, 

A sheet metal blank, B, for a templet, a 
bar of steel for a cutter-shaper, or a complete 
gear blank for a complete gear wheel, is 
fastened to the arc K. 

Now give the tool A a reciprocating motion, 
and slowly roll the blank B past it. An 
odontoid will be formed on the blank that is 
conjugate to the rack tooth, and, if it is 
formed of odontoids that are similar and 
symmetrical with respect to the pitch line C, 
all the odontoids made by it will be inter- 
changeable. 

The plane table, straight-edge and arc, 
slide and stand, can be accurately shaped by 
ordinary methods, but the shaping and plac- 
ing of the tool A requires considerable skill. 

The chief requirement is that the rack tooth 
shall be formed of four equal odontoids, 
a, 5, c, d, Fig. 14, placed symmetrically with 
respect to the pitch line P, and reversed with 
respect to the line of centers Q. If the four 
curves are odontoids, all the formed curves 
will be odontoids, but the set will not be 
mutually interchangeable unless they are also 




Conijugating tooth 

equal to each other and properly placed on 
the pitch and center lines. It is also desir- 
able, although not essential, that at their 
junction the curves should be tangent to the 
same straight line T. 

It is a matter of merely secondary practi- 
cal importance that the originating curves 
should conform to some exact predetermined 
shape, for as long as they are odontoids the 
system will be perfect, if formed by the 
oonjugator. If they are cycloids, the cycloidal 
system will be formed ; but if the cycloidal 
outline is imperfectly followed, the system 
may still be mechanically perfect. 

The tool A should be extended beyond the 
addendum line, as shown by dotted lines, so 
that the finished gear will have the usual 
clearance added to the working depth of the 
space. 

A rack tooth is chosen for the originating 
form on account of its simplicity. If the 
straight-edge C\b replaced by a circular arc, 
the flank curves of the tool A would not be 
like the face curves, and an interchangeable 
set could be obtained only by great skill in 
their formation. 

If the stand E tips a little out of a right 
angle with the table H, the cutter-shaper B 
will be formed with a clearance or " relief," 
and its deviation from a correct form will be 
slight. If the blank B is held on a slide that 
will move radially to the arc K while the cut 



THE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



is being made, a relief will be given to the 
shaper without injury to its form. 




INTEBNAI. GEAJRS. 

We have seen, Fig. 7, that similar odon- 
toids are conjugate and interchangeable, 
whether the pitch lines curve the same or 
opposite ways. 

When the two lines curve the same way, as 
in Fig. 15, the smaller gear will work inside 
the larger, which is then an internal gear. 

The theory and its application are, in the 
main, the same as with external gears, the 
only prominent distinction being the direction 
of the curvature of one of the lines. 

INTEBFEKENCE OF INTERNAL TEETH. 

With internal teeth we must guard against 
interference, for the face of the pinion is 
likely to interfere with the face of the gear in 
a certain position P . 

When the teeth come in contact at P', their 
common normal N' P' M' must pass through 
the pitch point 0. Drawing N' A and M' B, 



which will be parallel, we have Z M HP — 
AN" M' , and the normal angles are equal. As 
the two odontoids are similar, and the normal 
angles are equal, the normals P' N' and P' M 
are equal, and the point of contact bisects 
the chord M' N'. 

The normal of contact PO = P' N' =P' M' 
is therefore equal to c, the center distance 
A B, multiplied by the cosine of the normal 
angle F, or 

PO 
cos V 

and, if we draw the addendum at that value 
of P 0, the teeth will clear each other, 
just as they would otherwise interfere. If we 
know the form of the odontoid in use, we can 
express cos V in terms of P 0, and thence 
deduce the exact value of the latter that will 
let the teeth clear each other at the given 
center distance. 

In case the face or the flank odontoids are 
not similar, the system is still interchangeable 
to the extent that any pinion will work in any 
gear, and in that case, the requirement to 
avoid interference is that the sum of the nor- 
mals P JV' -^P' Jif = P0-{- Q must not 
be greater than M N'. 

LIMITING DIAMETEBS. 



Knowing the minimum value of 



P O 



for 



cos V 

any odontoid that may be in use, we can 

determine the least center distance between 

two gears that will work together, for that 

P O 
value of ^ is the required least center dis- 
cos V ^ 

tance. 



DOUBLE CONTACT. 



If 



PO 
COS V 



QO 
cos V 



is always equal to 2 c, the two faces will 
always be in driving contact, and, as the face 
of the pinion is also always in driving contact 
with the flank of the gear, we have a case of 
double contact as far as the truncation of the 
tooth will permit. 



THE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



O Q 

If - is given, we can assume a value for 

cos V ^ ' 



successive positions of the point of contact of 



an odontoid with the mating odontoid, 

P O As each normal, P K^ Fig. 19, comes to 

-^^^thatwillsatisfytherequirement,sothat ^^^ ^^^^^ p^^^^ ^^ ^^^ p^^^^ ^^ ^^^^^^^ p 

it is always ppssible to obtain double contact locates one point, P', of the line of contact, and 
by choosing special odontoids. 

Double contact is a curious but not a very 
valuable feature of gear teeth. 



Fig* 1^ Fig 




Fig. 18 



Cases of interference 

Internal gears may interfere from other 
causes, an odontoid from one crossing an 
odontoid from the other if the center distance 
is too small. 

For example, if the tooth of the 
pinion of Fig. 16 is cut down to the pitch 
line to avoid the ordinary interference, they 
may still interfere as shown, the flank of the 
pinion interfering with the face of the gear. 
Similarly if the gear face is cut down, the 
pinion face may interfere with the gear flanks, 
as shown by Fig. 17. 

The pinion face, when it cannot interfere 
with the gear face from the ordinary cause, 
may still cross it, as shown by Fig. 18. 

The only remedy for interference is to 
shorten the addendum, to increase the center 
distance, or to use odontoids that curve quick 
enough to pass. 



The 



LINE OF CONTACT. 

line of contact " is the locus of the 



Jjines of Contact 



the four similar odontoids of the complete 
tooth form the complete line of contact 
D' C. 

It generally takes the form of an hour- 
glass curve, is at right angles with the odon- 
toid at 0, and at right angles with the line of 
centers at D'. 

As all normals, on all similar odontoids, 
have the same length for the same normal 
angle, it follows that a system of any number 
of similar odontoids has a single and common 
line of contact, which has equal face and 
flank lobes. 



THE NORMAL THEORY OP THE GEAR TOOTH CTTRViiJ. 



ROLLED OUKVES. 

If a rolling curve or "roller," P K, be 
rolled on a pitch line, K, &, point, P, upon it 
will trace out a rolled curve, P. 

As the line P K, from the tracing point P 
to the point of contact K^ is always rotating, 
for the instant, about ^ as a center, the 
curve P will always be at right angles to it 
at P, and it is, therefore, always a normal to 
the curve. As each one of the normals is 
separate from the preceding and following, 
and the normal intersections with the pitch 
line are consecutive, the rolled curve is a true 
odontoid. As the length of the normal P K^ 
the normal angle P K 8, and the normal arc 
P jr= K, are in no way dependent upon 




When the tracing point is any ordinary 
point on the roller, the curve traced will be at 
right angles to the pitch curve, but when it 
is the pole of a spiral it may cross at an 
angle. Fig. 22. 

Although rolled curves and odontoids are 
identical, they cannot readily be considered 
the same, for the cycloid is the only 
odontoid worth noticing that can be con- 
veniently handled in shape of a rolled curve. 



Fig. 23 




Holled segment 



Rolled curve 



The involute can be formed by rolling a 
the curvature of the pitch line K, they are logarithmic spiral on the pitch line, but that 



the same for the same arc K on all curves, 
and therefore all odontoids traced on the 
same side of any number of different pitch- 
curves by the same point on the same roller 
are similar odontoids. 




Moiling 



feature is a mere curiosity without practical 
value, while the circular segmental tooth of 
Fig. 23, a perfect and very simple odontoid, 
can be formed only by a roller that is a 
curious combination of polar spirals that can 
be discussed only by the use of the higher 
mathematics. 

The properties of the odontoid can gen- 
erally be more easily developed and clearly 
explained if it is considered as a special case 
of the enveloping curve, than if it is treated 
as a rolled curve, while, for practical pur- 
poses, the conjugator, founded on the normal 
theory, has the advantage of any device that 
is founded on the rolled curve theory. 



APPLICATION OF THE THEORY 



PARTICULAR CASES 



\M 



Fig. 24: 

^ The Segmental System 




The Segmental System. — If a circular arc 
be drawn from a center A, Fig. 24, on a line 
at an angle, EGA, with a rack pitch line, 
O F, its normals P Kio the pitch line, will 
satisfy the law of the odontoid, and the seg- 
ment D F will be a rack tooth that will 
form an interchangeable or conjugate set of 
teeth. 

If the radius A is infinite, the segment 
is a straight line M N, b.\> right angles at 
with A, and the common involute system 
will bo formed. Therefore the involute tooth 
is a special form, the infinite form, of the 
segmental tooth. 



The segment has exactly the valuable prop- 
erties of the involute at the pitch line, and 
approximately away from it, the approxima- 
tion being closer as the radius A\b longer. 

The involute tooth is often, but not prop- 
erly, regarded as the special case of the 
cycloidal tooth for a rolling circle of infinite 
diameter. Kegarded simply as a curve, the 
involute is an infinite cycloid, but regarded 
as a gear tooth curve it is not, for, as shown 
by Fig. 37, infinite cycloids have a mathe- 
matical but not a practicable contact, and 
cannot bear properly, unless the conditions of 
the movement are so far strained that one is 



IHE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



reversed on its pitch line, and then the pitch 
lines are separated. 

FoBM OF THE SEGMENT Aii TooTH. — The face 
of any segmental tooth, on any circle with 
center at C, will be a lobe, d' f , where e' d' 
=e d, and ^ f —0 ef. It is always at right 
angles at with a. 

The flank curve D' F' is at right angles 
at 6> with A^ has the same height D' E\ 
and the same base E F' at the rack face. 

If a, O and A are in the same line, the face 
and flank will join at 0, and be a single 
curve. 



no cusp will be apparent, but the slightest 
increase of the proportion will separate the 
points. 

In the most convenient system, where 
A0E=1^° 28' 40", and sin. AOE=\, the 
cusps will appear whenever the segment 
and pinion radii are in the proportion 

^=:^/.i= 1.687. 
As the proportion 



OA 
00 



increases, the sec- 



ond branch (^ R will increase so that the curve 
will take the form OQ' B' D", and when OA 



Fig. 25 
Cusps of 

Segmental Flanks 




Cusps of Segmental Flank. — When the ra- 
dius J., Fig. 25, is small, compared with 
the radius OC, the pinion flank takes the 
form shown by Fig. 24; but as the pro- 

OA 
portion j^n increases , a value will be reached , 

OA 
when j^ = Y sin. AOE, at which a double 

cusp, Q' i?, will form. At exactly that point 
the two points Q' and R will coincide, and 



is infinite the second branch, Q' Z, then an 
involute, is infinite. 

The Segmental Delineatoe, — The seg- 
mental curve can be formed by the * * conjuga- 
tor" previously described, and shown by Fig. 
13, and it can be drawn by the special 
delineator, shown by Fig 26. 

A thin wooden wheel, C, turns on a pin at 
its center, and a rack, B, rolls on it, being 
held to it by a strip, aOc, of thin brass or 



THE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



strong paper attached to both. It is kept 
in position by a guide, H. 

A fixed bar O, projecting over the wheel, 
carries a pin 0, placed exactly at the point of 
contact of wheel and rack. A rule E turns 
about a pin A in the rack B, and carries a 
pointed tracing pin or pencil point at P, The 
pins A and P are in line, and all three are 
always at the same distance from the straight 
edge of the rule E. The pin P will pass 
undei and come in line with the pin 0. 

As the rack is rolled on the wheel, the rule 
will turn about the pin A and slide on the 
pin 0. The point P will trace segment of 



The action is practicable until the point of 
contact arrives at the first cusp Q' of Fig. 27 ; 
but beyond that, when it is on the second 
branch Q i?, the flank curve is inside the rack 
face, and the action is impracticable. 

There will also be an actual interference 
with the first branch. When the point of 
contact is on the second branch, the rack 
face will cross the first branch at J, and 
therefore the addendum must terminate the 
rack tooth at the point Q that conjugates 
with the cusp Q'. 

The difference between theoretical and 
practical contact is illustrated by the two ma- 




a circle P 8 with respect to the rack, but on 
the wheel will trace out the segmental flank 
0' Q R D'. 

If the pin A is carried by an arm on the 
other side of the rack pitch line, the face of 
the pinion tooth will be drawn, but, as the 
form of the face is very simple, the utility 
of the instrument is confined to the flank 
curve. 

Inteefeeence of Segmental Teeth. — The 
action of the segmental rack tooth on a flank 
that is conjugate to it, when the proportion is 
such that a cusp is formed, is always mathe- 
matically perfect, but not always practicable or 
capable of mechanical use. 



Fig, 26 
Segmental Delitieator 

chines, the conjugator of Fig. 13 and the de- 
lineator of Fig. 26. A full rack tooth on the 
conjugator will form the first branch correct- 
ly, but when the cusp is reached will return 
on it and cut it away, while the delineator, 
having but one acting point, will follow the 
theory and trace out all three branches. 

Least Numbee of Teeth. — Therefore, if the 
addendum is fixed, and it usually is, interfer- 
ence will generally set a limit to the diameter 
of the smallest pinion with which a rack tooth 
having the given addendum will work, with- 
out bearing on the second branch of the 
pinion flank. 

The diametral pitch being unity, a the ad- 



THE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



dendum EX, Fig. 25, 5, the segment radius 
OA, and F, the angle of obliquity AOE, the 
smallest possible number of teeth is 
2a sin V 



(l+sinv)' 



(1) 



If 5= 00 for the common involute system , 
a=l, and sin. V=.25, this formula gives t= 




Interference of 
Segmental Teeth 

32. Therefore, the common involute system 
cannot have an addendum of unity on an in- 
terchangeable set having gears with less than 
thirty -two teeth. When the set includes 12 
teeth, as is usual, the addendum must be 
shortened, or the points must be rounded 
over, as at QQ^fiT, Fig. 27. 

If we have given the angle of obliquity, 
the addendum, and the number of teeth in 
the smallest pinion, the largest possible seg- 
ment radius that can be used is 



h=- 



a 



V 



2a sin. Y 



—sin. V 



(2) 



This, for the common case, where «=1, i= 
12, and sin. F— .25, gives Z> = 10,34: as the ra- 
dius for the usual twelve-tooth system. A ra- 
dius of 13.91 will allow 15 teeth, 16.95 will 
allow 16.95 teeth, 23.58 will allow 20 teeth, 
and a short radius of 8.44 will admit a 10- 
tooth pinion. 

The Natubal Set. — There is one particular 
proportion of segment radius to pinion ra- 
dius, that might be considered the natural 
limit to the interchangeable system, and that 



is the proportion at which the cusp first ap- 
pears. If that, or a smaller proportion is 
chosen, there is no limit set to the addendum, 
and no interference is possible, for the second 
branch of the curve never appears. 

For that point we have the relation 

b=^ t sin. V, (3) 

so that, by choosing some value of t as the 
lower limit, we can find h for the whole set. 
If sin. F=.25, we find 5=f|«, giving &=8xV 
for a ten-tooth set, 5=10^ for a twelve-tooth 
set, 5=12|| for a fifteen-tooth set, 6=27 for a 
thirty-two-tooth set, and so on. 

If we use the plan previously explained, 
and calculate by formula (2), we can get a 
greater value for it, but in that case the ad- 
dendum is limited to its chosen value. 

As the addendum is always limited in prac- 
tice, almost always being unity, formula (2) 
appears to be better adapted to practical 
purposes than formula (3). 

CoBRECTED INVOLUTE TooTH. — We havc secu 
that the true involute tooth, when sin. V— 
.25, cannot be used for an interchangeable set 




Stunted Involute 

that includes gears with less than thirty -two 
teeth, if the addendum is unity. 

It is, however, customary to use the full 
addendum on a set that includes twelve teeth 
with the result that it must be corrected (?) 
for interference by rounding over the corners 
as in Fig. 27. 



THE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



Formnla (1) will apply to the involute if 



b=cc . In that case -r=o, and 



(4) 



If t=12, and sin. F=.25, we have a=f, so 
that the common involute, Fig. 28, is limited 
to the addendum «^=f , and the additional 
bc=^ of the full addendum must be cut off or 
got out of the way by rounding off as much 
of it as would interfere with a twelve-tooth 
pinion. 

This additional five- eighths is not inter- 
changeable, is not a tooth curve, and is kept 
on merely to give the appearance of a whole 
tooth. What usually appears to be a full ad- 
dendum, is really stunted to but little more 
than its third part. 




pitch 
line <* 



Action of 
Corrected Involute Tooth 

The only true correction, the only device 
that will allow of a full addendum of unity, 
retain the true involute for any part of it, and 
permit a rack to run in a pinion of less than 
thirty-two teeth, would be to correct the rack 
tooth by rounding over the point as in Fig. 
29, to give the flank the same correction, so 
that the condition of interchangeability is 
satisfied, and then to form a conjugate set 
from the corrected rack tooth. The result 
would be a mixed action : true involute, near 
the pitch line, and epicycloidal or otherwise 
away from it. 



This plan would have the serious defect that 
the corrected part bd must be a very defective 
odontoid, with a jerky and very nearly im- 
practicable action ; for, to obtain the neces- 
sary correction between b and d the curve 
must turn so quickly that its normal intersec- 
tions vrith the pitch line must be crowded 
within a narrow limit mn. 

Therefore, it would not appear to be advisable 
to correct the involute at all, for low-numbered 
pinions, but to discard it altogether, or to 
keep up appearances, as at present, by a 
merely ornamental and deceptive extension to 
nearly three times its effective length. 

If it is discarded, its valuable peculiarities 
will be lost, and therefore its substitute 
should be the curve that is nearest like it, and 
most nearly has its properties. 

Evidently, the nearest possible approach to 
the involute, is the segment that has the same 
angle of obliquity, and the longest radius 
that will admit the required addendum on 
the required smallest pinion, as found by 
formula (2). 

Figs. 29 and 30 serve to compare the action 
of the corrected involute with the segmental 
tooth. The action of the segment. Fig. 30, 
is exactly the same as that of the involute at 




pitch 
line 



Action of 
Segmental Tooth 

a, and its rapidity gradually increases to the 
finish at n. The corrected involute action, 
Fig. 29, is uniform from a to m, and finishes 
with a sudden jerk from mio n. 



THE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



If the correction should be curved enough 
to cause any of the normals to meet on or 
outside of the pitch line, the action would be 
wholly impracticable. 

The Ctcloldal System. — The cycloidal 
system is generally, but not properly, 
called the * ' epicycloidal " system. It is 
no more epicycloidal than it is hypocy- 
cloidal, for the faces are of the one form, 
and the flanks of the other. It is simpler and 
easier, as well as more correct, to apply the 
name cycloidal to both face and flank, and to 
the whole system, as is sometimes done. 

As before stated, the cycloidal system can 
be more easily developed and studied by the 
"rolled curve" theory than by the normal 



angles at and F, with a base, OF, equal in 
length to the circumference of the roller. 

As with all rolled curves, the line PE, from 
the tracing point tc the tangent point, is al- 

Fig. 31 




The Cycloid 

ways a normal to the curve, and therefore, to 
draw a normal to any given point P, strike 
a circle through the point having a diameter 



Fig. 32 
Cycloidal Teeth 




theory, because its roller, the circle, is the 
simplest of all curves. But, in this place, 
the former theory will not be used further 
than to define the nature of the cycloid, 
which is the generating odontoid that forms 
the system. 

The Cycloid. — If a circle A. Fig. 31. is? 
rolled on the straight line OF, a fixed point 
P, in it will trace out a transcendental curve 
OPDF, called the cycloid. 

It is a lobe, having a height, DE, equal to 
the diameter of the roller, and is at righ'. 



equal to the height DE, and tangent to the 
base line, and draw the normal PK to the 
point of tangency. 

As the arrangement of the normals is con- 
secutive, the curve is an odontoid, and all 
curves formed from it will be similar odon- 
toids that will work interchangeably with it. 

The simple process for drawing the normal 
makes it easy to form the conjugate face or 
flank belonging to any pitch circle. The 
flank cycloid Odf, Fig. 32, forms a face on 
the pinion, which is always a lobe Od' f bX 



THE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



right angles at with the pitch line, and 
meeting it again at/', where Oe f' = Oef. 

The face cycloid ODF forms a flank, 
OD' F\ on the pinion, that is at right angles 
with the pitch line at 0, meets it again at F\ 
where OD' F=ODF, and which takes vari- 
ous forms, according to the size of the pin- 
ion compared with that of the cycloid. 

When the radius 0(7, of the pinion, is 
greater than the height ED of the cycloid, 
the flank will be a concave lobe, OD' F' . 




When the radius OCis equal to the height 
ED, as in Fig. 33, the flank will be a straight 
diameter OF' . 




When the radius is less than the height, as 
in Fig. 34, the flank will be a convex lobe. 

As an undercurved flank, as in Fig. 34, is 
weak, it is customary to so limit the radius 
of the pinion that it shall never be less than 
the height of the originating cycloid. 

As the proportion of EDio OC still further 
increases, the flank is still more undercurved, 
until when 0C= ^ ED, we have the base, OF, 



equal to the circumference of the pinion ; and 
the flank is concentrated to a single point at 
0. The wearing action is also concentrated 
at the single point, and such a tooth, al- 
though practicable, is quite useless. 




If the height of the cycloid is greater than 
the diameter of the pinion, Fig. 35, the flank is 
a lobe, entirely external to the pitch line; and 
although the contact is still mathematically 
perfect, it is no longer practicable, for it is 
on the inside of the cycloid, as shown at P\ 




Cycloidal Involute 



If the proportion is carried to its extreme, 
the height being infinite as compared with 
the diameter of the pinion, as in Fig. 36, the 



THE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



cycloid becomes the straight line OD, and the 
pinion flank is the involute OD' . From this it 
is plain that the involute, in the form of an 
infinite cycloidal odontoid, is not a practi- 
cable gear tooth curve, the action between 
two gears being, as in Fig. 37, always on the 
straight line AOB at the crossing of the two 
tooth curves. 




Infinite Cycloidal Teeth 



CTOLOiDAii Lines of Contact. — The pri- 
mary lines of contact, in the case of the cycloid, 
are circles. Fig. 38, of the same diameters as 
the heights of the originating cycloids. 

The secondary lines of contact are also 
circles. The diameter is equal to the pitch 
diameter plus or minus the height of the 
cycloid. 



Inteenal Intereebenge. — With cycloidal 
teeth we cannot have a partial interference, 
as with segmental teeth, which can be 
remedied by truncation of the teeth, for 
the exterior secondary of the pinion is either 
entirely inside the interior secondary of the 
gear or entirely outside of it, except in 



ence between the pitch diameters is greater 
than the sum of the heights of the originat- 
ing cycloids, there can be no interference, 
but if it is less there will be a continual inter- 
ference that can be remedied only by the en- 
tire removal of the face of one of the teeth. 
The condition of non-interference can be 
conveniently expressed by the rule that the 
difference between the numbers of teeth on 
the gears must not be less than the half sum 
of the numbers of teeth on the base gears. 
This, for the common interchangeable sys- 
tem, requires that there should be a differ- 
ence between the gears at least as large as the 
base gear. For example, in the fifteen tooth 
set there must be at least fifteen more teeth 
in the gear than in the pinion.* 



Double Internal Contact. — When the 
condition of non-interference is exactly satis- 
fied, there is a case of double contact, for 
then the two secondaries coincide. In a case 
of double contact of interchangeable teeth, 
the coinciding secondaries must exactly bisect 
the chord c d of Fig. 20, for the primaries are 
equal, and their chord 6 / is exactly bisected 
in that case. As the circle is the only curve 
that will bisect all the chords c d, it follows 
that the cycloidal system is the only one that 
can have double contact, and at the same time 
be interchangeable. 



Pbactical Construction. — The practical 
application of the conjugating process, in the 
case of the cycloid, presents the difficulty 
that the originating rack tooth must be an 
exact cycloid, which condition can be mei 
only by special mechanism. 

The originating segmental tooth has an 
outline that is formed of arcs of circles, and 
that of the involute is composed of straight 
lines, and both can be easily shaped. But 
the difficulty in the practical application of 
the cycloidal system is not by any means the 
greatest objection to it in comparison with 
its simpler and superior rival. 



the case of entire coincidence. If the differ 

* The discovery of the law of internal interference, as far as it relates to cycloidal teeth, is generally 
credited to Prof. C. W. MacCord ; but, in claiming that discovery, the professor could not have been aware 
of its previous publication, by A. K. Mansfield, in the Journal of the Franklin Institute for Januaiy, 1877. 



THE NORMAL THEORY OF THE GEAR TOOTH CURVE. 



Lines of Contact 




The comparative EFFICIENCY of the TEETH of 

GEARS. 



[Reprinted from the Journal of the Franklin Institute, May, 1887.] 

The effect of friction between the teeth of gears is not well 
understood, and the popular impression, even among educated 
engineers, concerning the comparative efficiency of the two forms 
of teeth in common use — the involute and the cycloidal — is that 
the latter is much the most economical, and, therefore, much better 
adapted for use for the transmission of heavy power. 

This impression is entirely wrong, the reverse of the provable 
facts, and it is based not entirely on fancy but partly on the 
teaching of authorities that are undoubtedly competent. 

It is with no small feeling of timidity, that I venture to contra- 
dict the declared and apparently proved opinions of such high 
authorities as Reuleaux, Herrmann and others, and I would not 
dare to assert a contrary view if I did not feel able to prove it, by 
evidence that will bear the closest examination. I will give the 
demonstration in great detail, so that it can be followed by any 
one who is familiar with the common processes of analysis. 

By the work done by a gear wheel, I mean the work done by 
the friction of sliding between the teeth. I shall leave out the 
small rolling friction between the teeth, and I ' shall not consider 
the friction of the shaft bearings. 

The work lost by the rubbing of two surfaces on each other is 
the product of the normal force acting between the two surfaces, 
by the distance through which the resistance is overcome, and by 
the coefficient of friction for the material in use. 

To determine the work done by a pair of gear teeth, we must 
determine these three factors or their product, and this may be 
done in two different ways : by a graphical process, and by an 
analytical method. The two processes are entirely independent of 
each other beyond the given premises, and their agreement upon 
a common result is a substantial proof of the accuracy of both. 

Graphical Process. — In Fig, J, the two tooth curves have rubbed 
upon each other, while the point of contact between them has 
moved from C to ^ on the line of action, AD, and they have 



done work that is the product of the coefficient of friction,/, by 
the difference, zy, of the lengths of the curves that have passed the 
point of contact, and, for graphical purposes, of the average force, 
/S", that has acted between the two teeth. 

If we make a drawing, showing the two teeth in several posi- 
tions, preferably at equal intervals of their action, we can deter- 
mine the work done within the limits of each interval by multi- 
plying together the factors as found by measurement. The total 
work done between any two points is the sum of these products 
for all the intervals between the points. 

In Fig. 2, this process is applied to an exaggerated example of 
a pair of cycloidal teeth. The gears, with radii h and hy have ten 




Fig. I. Analytical Process. 

and twenty teeth, the tangential force, P, between the two gears 
is assumed to be constant and unity, and the coefficient of friction 
is assumed to be one-tenth. The describing circle, with radius 
0M= Ty has three teeth, so that a gear of six teeth would have 
radial flanks and be the base or smallest gear of the interchange- 
able set to which the two gears belong. 

The pitch and describing circles are divided into equal inter- 
vals, Oa, ah, he, etc., of one-twelfth of the whole tooth arc, or cir- 
cular pitch, Olj commencing at the line of centres, and the v/ork 
done over each of these small intervals is to be determined. 

Make a templet of an epicycloid on the gear h, and of a hypo- 
cycloid within the gear h, and draw curves from each of the divi- 
sions of the pitch lines. Each pair of curves should meet on the 



corresponding division of the describing circle. Measure the dif- 
ferences between the lengths of these curves (see column 2 of the 
table), and by subtracting each total difference from the next 
larger, find the partial length of curve passed over during each 
interval, as tabulated at column 3. 

Draw a line at an arbitrary distance, representing unity, from the 
line of centres and parallel with it, and draw lines, OSaj OSb, OSc, 
etc., through the centres of the intervals. The length of each line 
(column 4) can, with small error, be assumed to be the average 
normal force for its interval. These normal forces can be very 




Fig. 3. Graphical Process. 

easily computed, for each one is the reciprocal of the cosine of the 
angle POS. The angle for the first normal is 2^°, and there are 
5° between each of the following normals: 

Multiplying together the normal for each interval, the partial 
curve for that interval, and the coefificient of friction, we obtain 
the loss for each interval as tabulated at column 5. By summa- 
tion we obtain the total loss to and including each interval, as 
tabulated at column 6. 

For the involute tooth, we have a constant normal force, ^=r 1*15, 
the total work done, column 10, up to any interval is the product 



of that force by the total curve, column 9, for that interval. The 
figure is so similar to Fig. 2^ that it need not be given here. 

The graphical process will determine the general result, and 
show that while the two curves are substantially equal in efficiency, 
the advantage is a very little in favor of the involute. If we wish 
a precise comparison between these two curves, no graphical 
process can be used, and we must resort to analysis. 



Analytical Process, — In Ftg. /, the two tooth curves are odon- 
toids of any possible form, and they will secure a uniform velocity 
ratio between the pitch lines. They slide on each other, the point of 
contact moving along the line of action, AD. At any time they 
are at a distance A = h from the pitch point, 0, and are pressed 
together with a normal force, St which is equal to the constant 
tangential force, P, divided by the cosine of the angle of obliquity, 
P A= V, and this normal force is always in the direction of the 
pitch point 0. 

While the normal, O A, turns through an elementary angle, the 
arc of which is d V, the two curves will rub on each other over an 
elementary distance, A B = A ' d V=b ' d V, and they will 
do the elementary work 

dW = fP'AB'S'==f-^'b'd V. 

cos V 

At the same time the wheel h will turn through the elementary 
angle, the arc of which is 

dx = -^dV 

in which the positive sign is for external, and the negative sign is 
for internal contact. 

Therefore, we have the total work done by friction, while the 
wheel h is turning through an angle, the arc of which is x. 



TTT f p h ± h rb dx 



k •^^cos V 
o 

and this cannot be carried further until we know the form of tooth 
curve to be used, and can determine b and cos Fin terms of x. 
First take the involute tooth. 



The distance ^ = 6 is equal to hx . cos V, and we have the 
total work done 

z 

I = fF.t^hfxdx, 

o 

which integrates to 

or, if we use the arc on the pitch line, w = hx,we have 

1 = ^-_ . — — — w^ 
2 kh 

for the value of the work done by the friction of involute teeth 
while moving from the pitch point over any arc, v), on the pitch 
circle. 

It is a singular fact that this loss of power is the same for all 
values of the angle of obliquity. All involute systems are equal 
in efficiency, without regard to the angle of obliquity. 

Then take the cycloidal tooth. 

We have b = 2 r . sin — x, and cos V = cos — x, giving 
2r ' 2r ' 

the total work. 

X 

E=fP.^i^2rCian ^dx, 
•^ kh ^^ 2r ' 

o 



which integrates to 

js; = — 

the value of the total work of a pair of cycloidal teeth. 



E = — /P . ^r/ 4 r^ nat log cos ^, 
•^ kh ^ 2r 



To compare the cycloidal with the involute tooth for the same 
arc of action from the pitch point, divide E by /. 

Sr^ nat log cos — - 
E 2 V' 

T w" 

As this is unity for w = and greater than unity for any 
finite value of w, it follows that the efficiency of the involute is 
mathematically superior to that of the cycloidal curve, in all cases 
and under all circunistances, without regard either to the angle of 
obliquity of the involute, the size of the describing circle of the 
cycloidal curve, or the arc of action, and provided only that the 



comparison is made over the same arc of action. (See column 
13 of the table.) 

In both of these formulae it is seen that h and h can exchange 
places without affecting the result for external contact, and there- 
fore the work done is the same, for the same arc of action, on both 
sides of the line of centres, the tangential force being constant. 



For a comparison between external and internal gears, we have 
Ji. ^ lor E Ext ^ h ^h 
B Im-EInt. k — h 
so that the internal gear is much the most economical, particularly 
when the two gears are nearly of the same size. 

When k = 2h WQ have -g- = 3. That is, if the internal gear 

is twice the size of its pinion, the work lost is but one-third of that 
lost when both gears are external. 

Small improvement can be made by putting a small pinion in- 
side, rather than outside of a large gear, as is often done at great 
expense on boring mills and large face plate lathes. A six-inch 

pinion and a six-foot gear will give -^ = 1-18 an advantage of no 

Jo 

great value. 



It is seen from the above that the work being done increases 
very rapidly with the arc of action ; with the square of that arc in 
the case of the involute, and in a still greater proportion for 
cycloidal teeth, and hence that arc should always be made as 
small as possible. 

Strength should be secured by a wide face rather than by a 
large tooth, for the face of the gear has no influence on its effi- 
ciency. 



The two formulae for E and Jean be very easily applied to any 
particular example, and the results obtained much more quickly, 
as well as more accurately than by the graphical method. 

For application to the given example, where h = 10, k = 6y 
y = 1, and P = 1, we have 

E = 6-21 70 [0 — log cos (5 n)^] 
I = -01028 n^ 
in which n is the number of any interval, C, is the characteristic 



with the sign changed, and hg cos contains only the mantissa of 
the common logarithmic cosine of 5 n^. 



It is seen from the tabulated value of E and J obtained by com- 
putation, columns 7 and 11, that the graphical and analytical pro- 
cesses agree very closely, the errors being shown by columns 
8 and 12. As before stated, this agreement is a strong indication 
of the accuracy of both. 

Prof. Reuleaux* finds that the two curves are exactly equal 
when compared over the same arc of action, and Prof. Hermannf 
finds the same result by a different process. In both cases the 
result was arrived at by making an approximation, for reasons not 
given but probably to simplify the work. 

If the actual determination of the work done is the end in view, 
the approximations can be allowed, as the result is then close 
enough for all practical purposes. But, if the object is a close com- 
parison between the two curves, the slightest difference must be 
accounted for, and neither Reuleaux's nor Herrmann's formulae will 
answer the purpose. 

Herrmann remarks, *< It is evident, moreover, that the friction 
of involute teeth will be somewhat greater than that of cycloidal 
teeth, the angle y being smaller for the former than for the latter." 

This may be " evident," but it is not provable, and the state- 
ment that the angle y, which is the complement of the angle of 
obliquity, is smaller for the involute, is not correct. Up to the half 
tooth point it is so, but beyond that point the reverse is true. At 
the half tooth point the two forms always have the same angle of 
obliquity if they belong to interchangeable sets which have the 
same base gear. 

Further, it does not follow that the work of friction is the greater 
when the angle of obliquity is the greater, for the work of friction 
depends on two variable factors, the normal pressure, which indeed 
increases with that angle, and the length of the curve that is rubbed 
over. Within the half tooth point this curve is the shortest for 
the involute, so that the work done is the smallest although the 
other factor is the greatest. 

* Transactions of the American Society of Mechanical Engineers , vol. viii, 
1886. The result, without the demonstration, is also given in Reuleav^'s Kon- 
sirukteur, § 213. 

t Klein's translation of Herrmann's revision of Weisbach's Mechanics of 
Engineering and Machinery ^ vol. iii, § 79. 



As Herrmann states, " This difference is insignificant for the 
tooth profiles ordinarily employed,'* but the general impression, 
which it is the object of this paper to contradict, is that the differ- 
ence is very significant and in favor of the cycloidal tooth. 

Reuleaux goes further, and, after finding that the two curves are 
exactly the same for the same arc of action, gives several practical 
examples, which show the involute to be decidedly inferior, the 
difference being from sixty to eighty per cent. 

This result is correct for the conditions of Reuleaux's examples, 
but it seems to me that those conditions are not correct if the object 
is to compare the two curves, for he does not take them on the 
same terms. He takes the involute with a long arc of action, and 
compares it with a cycloidal tooth having a short arc, and of course 
the involute is then inferior. 



Example for h - 


= lo- 


/& = 


= 5- r 


= 1-5 /= 


= -I 


AND P = I 




^ 


Cycloidal Tbkth. 


Involute Teeth. 
Obliquity, 30°. S= 1-15. 




i 


Total 
Curve 


Par- 
tial 
Curve 


Normal 
Force. 


Par- 

tial 
Work 


Toial Work. 


Total 
Curve 


Total Work. 




g 


Graph 


Anal's 


Error 


Graph 


Anal's 


Error. 


E 

I 


I 


•oio 


•010 


1-0009 


-0010 


-ooii 


•00103 


•0001 


•02 


•0023 


•00103 


•0013 


1-002 


2 


•035 


•025 


1*0087 


-0025 


•0036 


-00413 


•0005 


•04 


-0046 


-0041 1 


•0005 


I 004 


3 


•085 


•050 


10243 


-0510 


•0086 


•00937 


-0008 


-08 


•0092 


•00924 





I -013 


4 


•155 


•070 


1-0485 


•0735 


-0160 


-01679 


-0008 


•15 


•0173 


-01645 


-0008 


I -021 


5 


•245 


•090 


1-0824 


•0975 


-0257 


•02656 


•0009 


•22 


0254 


•02570 


•0003 


1033 


6 


■355 


•no 


1-1274 


•1240 


•0382 


•03884 


•0006 


•32 


•0370 


•03701 





1-049 


7 


•485 


•130 


1.1857 


•1540 


-0536 


•05386 


•0003 


•44 


•0508 


•05038 


•0004 


i^o69 


8 


•630 


•145 


1-2604 


•182s 


•0718 


•07196 


•0002 


•57 


•0658 


-06580 





1-094 


9 


•790 


•160 


1-3563 


•2170 


•0935 


•09357 


•0001 


•72 


•0831 


•08328 


•0002 


I -124 


lO 


•965 


•175 


1-4802 


•2590 


•I 194 


•11932 


-oooi 


•90 


•1039 


•10281 


•oon 


1-161 


ZI 


X-150 


•185 


1-6426 


-3040 


•1498 


•15008 


•0003 


x-09 


•1259 


•12440 


•0015 


1206 


Z2 


1-350 


•195 


1-8615 


•3630 


•1861 


•18715 


•001 1 


i^30 


-1501 


•14805 


-0020 


1-264 


' 


* 


3 


4 


5 


6 


7 


8 


9 


10 


zz 


Z2 


X3 



The work done increases rapidly with the distance of the point 
of contact from the line of centres, and the result of Reuleaux's 
method is to compare one curve that is at work a considerable dis- 
tance from the line with another that is nearer to it. 

This is clearly shown by the figures of Reuleaux's comparative 
examples, for in each case the losses are almost exactly proportional 
to the arcs of action. 



For the purpose of comparison, the two teeth should be taken 
under precisely the same circumstances, and they should commence 
work and stop work together. They should have the same arc of 
action rather than the same addendum, for the addendum has very 
little to do with the gear except by its effect on the maximum arc 
of action. 

When taken under similar circumstances ^ involute and cycloidal 
gear teeth are practically equal with regard to the work done by fric 
tion, the difference being always slightly in favor of the involute. 



THE LIMITING NUMBERS OF GEAR TEETH. 

The treatment of the subject of the limiting numbers of gear teeth is usually 
so difficult that the student is obliged to take the results as he finds them ; for 
it is a great work of time and patience to follow out the process, and prove the 
results to be either true or false. 

The following processes are easily derived from the trigonometrical condi- 
tions of the problem, but I will here give the results only.* 

Assume tbe arc of recess to be a times, and the thickness of the tooth to be 
b times the circular pitch, and the diametral pitch to be unity. Let d be the 
number of teeth in the driver, and/ the number in the follower. 

For the cycloidal system, assume the diameter of the describing circle to 
be q times the diameter of the follower, and the limiting numbers of teeth 
will be involved in the following equation: — 

fQ 1 

d 



. r360° / b\ , 360° a n 



(1) 



. 360° 
sin 



^0°/ b \ 

d^V—r) 



which is insoluble in general terms, but from which either/ or q can easily be 
separated, for any particular case, by a few numerical trials. 

For a common example, assume the driver to have six teeth, the arc of re- 
cess to be equal to the pitch, the tooth to be equal to the space, and the flanks 
of the follower to be radial. This gives a = l, b — ^, q — J, and d = 6; so 
Miat the formula becomes 

f= ' ^^ 

20° ^ (2) 



) ~? 



To solve this, put / equal to two numbers as near truth as can be estimated, 
say 140 and 160. This gives 140 = 140.171, and 160 = 159.193, the opposite 
errors showing that / is between the two chosen points. 

Intei-polating in proportion to the two errors, we get 143.5 as our first 
approximation. 

Trying 143 and 144 in the same way, we get 143.491 as a second approxima- 
tion, and 144 as the required nearest larger integer. 

If the chosen points had been 120 and 180, the first approximation would have 
been 144.5, and a second trial would have fixed 144 as the nearest integer. 

When the driver is a rack, we must use the formula 



Z TT 



(^-^) 



^ . 360° a ^^^ 

q sm — r- ■ — 



and when the rack is driven we must use 

o 



d 



360-/ h \ 

which are simpler than the unlimited formula. 

*This subject I have treated in full, with illustrations and examples, in a paper in the 
bcientihc American Supplement, Vol. XXIII., 1887. 



When the involute system is to be treated, the problem is a double one; 
for the action on one side of the line of centers will set one limit, while that 
on the other side will set another. 

If we know Q, the angle of obliquity, we have 

/=2«7rcot^ (6) 

so that the problem is reduced to finding the value of Q for the given condi- 
tions. 

The solution is exact, and not dependent, as with cycloidal teeth, on a pro- 
cess of trial and error. 

On the approach side we have the formula 



tan 



« = ^0-'^) (6) 



and on the recess side the formula 



cos Q = / j>cot TT-f ^ -f V^;^ cot W-p'-i-i ^^^ 

in which P=2^ (8) and F = 2^ (a - -|-) (9) 



The approach will set a minimum value for Q, and the recess will determine 
a maximum. The maximum must evidently be no less than the minimum. 

When the involute rack follows, we have the same case as for a cycloidal 
pinion and rack, see (4) ; but when the rack drives we can use 



cos ^ = ^ 1 _ ' (10) 



The direct solution is somewhat tedious in application, and may be simplified 
by the use, on the recess side, of the formula 

^ p sin W ,,_ 

cos Q = — ,^, , — ^j7- (11) 

^ cos {Q-{- W) ^ ' 

which can be easily worked by the above-described process of trial and error. 

This supposes the involute to be for the interchangeable system, but when 
it can be allowed to be non-interchangeable the angle of obliquity on the 
approach need not be the same as that on the recess. The interchangeable 
involute tooth will not permit as small pinions as the non-interchangeable 
cycloidal tooth, but when both forms are taken on the same terms, both non- 
interchangeable, the advantage of the cycloidal tooth is destroyed. 



CONIC PITCH LINES. 



The utility of the conic sections, used as the pitch lines of gear wheels, lies 
/n the fact that under certain conditions they will roll together in perfect 
rolling contact when mounted upon tlxed centers. 



We can put all the conic sections under one law as to each of several fea- 
tures when rolling together, as follows : 

Any two equal conic sections will roll together in perfect rolling contact 
when fixed on centers at their opposite foci. 

Their moving foci will move at a fixed distance apart. 

The two curves will make a continuous and complete revolution on each 
other. 

The point of contact of the the two curves will be at the intersection of the 
line of the fixed foci with the line of the moving foci. 

The common tangent to the two curves at their point of contact will pass 
through the point of intersection of the two axes. 



There are four conic sections, varying principally as to their focal distance. 
The circle, having an infinitely small focal distance : the ellipse, having a finite 
and positive focal distance; the parabola, having an infinitely great focal 
distance ; and the hyperbola, having a finite and negative focal distance. 

Any two curves that will roll together may be used as the pitch lines of 
gear wheels, and therefore we can have gears with either circular, elliptic, 
parabolic, or hj'perbolic pitch lines. 

In either case the moving foci may be connected by a link that will hold 
the two gears together when in motion, and this link will act in the most di- 
rect and advantageous manner when most needed, when the action of the 
teeth becomes so oblique as to be of little service. 



The four cases are illustrated by the four figures 




Fig. I 
CIRCULAR GEAP.S 

Case I. — When the focal distance is infinitely small the curves are circles, 
as in figure I. The lipk is here simply a fixed bar connecting the two centers, 
for the two foci are combined in one point at the center. 



ving focus 

Fig. IT 
ELLIPTIC GEARS 




Case IT. "When the focal distance is finite and positive, the curves are 
ellipses that will roll together if fixed on centers at their opposite foci, as in 
Fig. II. The link is a moving bar connecting the two moving foci. 

/ 




Us ICig. Ill 

JPAHABOLIC GEARS 



Case III. When the focal distance is infinitely great the curves are parab- 
olas. One parabola turns about its focus while the other turns about its 
opposite focus, but, as the opposite focus is at an infinite distance the sec- 
ond parabola must move in a straight line at right angles to the line of centers, 
as in Fig III. The link becomes a bar of infinite length, and cannot be prac- 
tically applied. The revolution is complete but of infinite extent, so that it 
cannot be practically accomplished. 




Case IV. Wlien the focal distance is finite and negative, the curves are 
hyperbolas. The opposite focus about which one hyperbola turns is now on 
tlie other side of the curve, which becomes a negative or internal pitch line, 
as in Fig IV. The linli is of finite length and can be practically applied. 
Tlie revolution is complete, for as soon as one pair of curves separate, the 
other pair come together, and the motion is continued. 



The utility of circular gears is universal, and elliptic gears have many 
applications, but no use is apparent for parabolic or hyperbolic gears. A use 
for them will probably be found when their existence and properties become 
well known, and they are certainly cf interest to the student of mechanism' 



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